EIGENVALUE ESTIMATES FOR NON-NORMAL MATRICES AND THE ZEROS OF RANDOM ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

E. B. DAVIES^{1} AND BARRY SIMON^{2}

Abstract. We prove that for any n × n matrix, A, and z with |z| ≥ ∥A∥, we have that ∥(z − A)^{−1}∥ ≤ cot(\frac{π}{4n})dist(z, spec(A))^{−1}. We apply this result to the study of random orthogonal polynomials on the unit circle.

1. Introduction

This paper concerns a sharp bound on the approximation of eigenvalues of general non-normal matrices that we found in a study of the zeros of orthogonal polynomials. We begin with a brief discussion of the motivating problem, which we return to in Section 7.

Given a probability measure dμ on ℂ with

∫ |z|^{n} dμ(z) < ∞  (1.1)

we define the monic orthogonal polynomials, Φ_{n}(z), by

Φ_{n}(z) = z^{n} + lower order  (1.2)
∫ \overline{z^{j}} Φ_{n}(z) dμ(z) = 0 j = 0, 1, …, n − 1  (1.3)

If

P_{n} = orthogonal projection in L^{2}(ℂ, dμ)

onto polynomials of degree n − 1 or less

(1.4)

then

Φ_{n} = (1 − P_{n})z^{n}  (1.5)

Date: September 1, 2005.

1 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom. E-mail: E.Brian.Davies@kcl.ac.uk. Supported in part by EPSRC grant GR/R81756.

2 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.

E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

A key role is played by the operator

A_{n} = P_{n}M_{z}P_{n} ↾ Ran(P_{n})  (1.6)

where M_{z} is the operator of multiplication by z and A_{n} is an operator on the n-dimensional space Ran(P_{n}).

If z_{0} is a zero of Φ_{n}(z) of order k, then f_{z_{0}} ≡ (z − z_{0})^{−k}Φ_{n}(z) is in Ran(P_{n}) and

(A_{n} − z_{0})^{k}f_{z_{0}} = 0 (A − z_{0})^{k−1}f_{z_{0}} ≠ 0  (1.7)

which implies

Φ_{n}(z) = det(z − A_{n})  (1.8)

Also, Φ_{n}(z) is the minimal polynomial for A_{n}.

In the study of orthogonal polynomials on the real line (OPRL), a key role is played by the fact that for any y ∈ Ran(P_{n}) with ∥y∥_{L^{2}} = 1,

dist(z_{0}, {zeros of Φ_{n}}) ≤ ∥(A_{n} − z_{0})y∥ (OPRL case)  (1.9)

This holds because, in the OPRL case, A_{n} is self-adjoint. Indeed, for any normal operator, B, (throughout ∥ · ∥ is a Hilbert space norm; for n × n matrices, the usual matrix norm induced by the Euclidean inner product)

dist(z_{0}, spec(B)) = ∥(B − z_{0})^{−1}∥^{−1}  (1.10)

and, of course, for any operator C,

inf{∥Cy∥ | ∥y∥ = 1} = ∥C^{−1}∥^{−1}  (1.11)

We were motivated by seeking a replacement of (1.9) in a case where A_{n} is non-normal. Indeed, we had a specific situation of orthogonal polynomials on the unit circle (OPUC; see [16, 17]) where one has z_{n} ∈ ∂𝔻 = {z | |z| = 1} and unit trial vectors, y_{n}, so that

∥(A_{n} − z_{n})y_{n}∥ ≤ C_{1}e^{−C_{2}n}  (1.12)

with C_{2} > 0. We would like to conclude that Φ_{n}(z) has zeros near z_{n}.

It is certainly not sufficient that ∥(A_{n} − z_{n})y_{n}∥ → 0. For the case dμ(z) = dθ/2π has Φ_{n}(z) = dist(1, spec(A_{n})) = 1, but if y_{n} = (1 + z + ⋯ + z^{n−1})/\sqrt{n}, then ∥(A_{n} − 1)y_{n}∥ = ∥P_{n}(z − 1)y_{n}∥ = n^{−1/2}∥P_{n}(z^{n} − 1)∥ = n^{−1/2}∥1∥ = n^{−1/2}. As we will see later, by a clever choice of y_{n}, one can even get trial vectors with ∥(A_{n} − 1)y_{n}∥ = O(n^{−1}).

Of course, by (1.11), we are really seeking some kind of bound relating ∥(A_{n} − z_{n})^{−1}∥ to dist(z_{n}, spec(A_{n})). At first sight, the prognosis for this does not seem hopeful. The n × n matrix,

N_{n} = (\begin{matrix}0 & 1 & & 0 \\ & ... & ... \\ & & ... & 1 \\ 0 & & & 0 \end{matrix})  (1.13)

has

∥(z − N_{n})^{−1}∥ ≥ |z|^{−n}  (1.14)

since (z−N_{n})^{−1} = ∑_{j=0}^{n−1} z^{−j−1}(N_{n})^{j} has z^{−n} in the 1, n position. Thus, as is well known, ∥(A_{n} − z)^{−1}∥ for general n × n matrices A_{n} and general z cannot be bounded by better than dist(z, spec(A_{n}))^{−n}. Indeed, the existence of such bounds by Henrici [4] is part of an extensive literature on general variational bounds on eigenvalues. Translated to a variational bound, this would give dist(z_{n}, {zeros of Φ_{n}}) ≤ C∥(A_{n} − z_{n})y∥^{1/n}, which would not give anything useful from (1.12).

We note that as n → ∞, there can be difficulties even if z_{0} stays away from spec(A_{n}). For, by (1.14),

∥(1 − 2N_{n})^{−1}∥ ≥ 2^{n−1}  (1.15)

diverges as n → ∞ even though ∥2N_{n}∥ is bounded in n.

Despite these initial negative indications, we have found a linear variational principle that lets us get information from (1.12). The key realization is that z_{n} and ∥A_{n}∥ are not general. Indeed,

|z_{n}| = ∥A_{n}∥ = 1  (1.16)

It is not a new result that a linear bound holds in the generality we discuss. In [11], Nikolski presents a general method for estimating norms of inverses in terms of minimal polynomials (see the proof of Lemma 3.2 of [11]) that is related to our argument in Subsection 6A. His ideas yield a linear bound but not with the optimal constant we find.

Our main theorem is

Theorem 1. Let M_{n} be the set of pairs (A, z) where A is an n × n matrix, z ∈ ℂ with

|z| ≥ ∥A∥  (1.17)

and

z ∉ spec(A)  (1.18)

Then

c(n) ≡ sup_{M_{n}} dist(z, spec(A))∥(A − z)^{−1}∥ = cot(\frac{π}{4n})  (1.19)

Of course, the remarkable fact, given (1.14), is that c(n) < ∞ when we only use the first power of dist(z, spec(A)). It implies that so long as (1.17) holds,

dist(z, spec(A)) ≤ c(n)∥(A − z)y∥  (1.20)

for any unit vector y. For this to be useful in the context of (1.12), we need only mild growth conditions on c(n); see (1.21) below.

As an amusing aside, we note that

c(1) = 1 = 0 + \sqrt{1}
c(2) = 1 + \sqrt{2}
c(3) = 2 + \sqrt{3}

but the obvious extrapolation from this fails. Instead, because of properties of cot(x),

c(n) ≤ \frac{4}{π} n  (1.21)
\frac{c(n)}{n} is monotone increasing to \frac{4}{π}

so, in fact, for n ≥ 3,

\frac{2 + \sqrt{3}}{3} ≤ \frac{c(n)}{n} ≤ \frac{4}{π}

a spread of 2.3%.

We note that, by replacing A by A/z and z by 1, it suffices to prove

sup_{∥A∥<1} dist(1, spec(A))∥(1 − A)^{−1}∥ = cot(\frac{π}{4n})  (1.22)

and it is this that we will establish by proving three statements. We will use the special n × n matrix

M_{n} = (\begin{matrix}1 & 2 & … & 2 \\ 0 & 1 & … & 2 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \end{matrix})  (1.23)

given by

(M_{n})_{kℓ} = \begin{cases}2 & if k < ℓ \\ 1 & if k = ℓ \\ 0 & if k > ℓ \end{cases}

Our three sub-results are

Theorem 2. ∥M_{n}∥ = cot(π/4n)

Theorem 3. For each 0 < a < 1, there exist n × n matrices A_{n}(a) with

∥A_{n}(a)∥ ≤ 1  spec(A) = {a}  (1.24)

and

lim_{a↑1} (1 − a)(1 − A_{n}(a))^{−1} = M_{n}  (1.25)

Theorem 4. Let A be an upper triangular matrix with ∥A∥ ≤ 1 and 1 ∉ spec(A). Then

dist(1, spec(A))|(1 − A)_{kℓ}^{−1}| ≤ \begin{cases}2 & if k < ℓ \\ 1 & if k = ℓ \\ 0 & if k > ℓ \end{cases}  (1.26)

Proof that Theorems 2–4 ⇒ Theorem 1. Any matrix has an orthonormal basis in which it is upper triangular: One constructs such a Schur basis by applying Gram-Schmidt to any algebraic basis in which A has Jordan normal form. In such a basis, (1.26) says that

dist(1, spec(A))∥(1 − A)^{−1}y∥ ≤ ∥M_{n}y∥ ≤ ∥M_{n}∥ ∥y∥

so Theorem 2 implies LHS of (1.22) ≤ cot(π/4n).

On the other hand, using A_{n}(a) in dist(1, spec(A))∥(1−A)^{−1}∥ implies LHS of (1.22) ≥ cot(π/4n). We thus have (1.22) and, as noted, this implies (1.19). □

To place Theorem 1 in context, we note that if |z| > ∥A∥,

∥(z − A)^{−1}∥ ≤ ∑_{j=0}^{∞} |z|^{−j−1}∥A∥^{j} = (|z| − ∥A∥)^{−1}  (1.27)

So (1.19) provides a borderline between the dimension-independent bound (1.27) for |z| > ∥A∥ and the exponential growth that may happen if |z| < ∥A∥, essentially the phenomenon of pseudospectra which is well documented in [23]; see also [14].

The structure of this paper is as follows. In Section 2, we will prove Theorem 4, the most significant result in this paper since it implies c(n) < ∞ and, indeed, with no effort that c(n) ≤ 2n. Our initial proofs of c(n) < ∞ were more involved — the fact that our final proof is quite simple should not obscure the fact that c(n) < ∞ is a result we find both surprising and deep.

In Section 3, we use upper triangular Toeplitz matrices to construct A_{n}(a) and prove Theorem 3. Sections 4 and 5 prove Theorem 2; indeed, we also find that if

(Q_{n}(a))_{kℓ} = \begin{cases}1 & if k < ℓ \\ a & if k = ℓ \\ 0 & if k > ℓ \end{cases}  (1.28)

then

∥Q_{n}(1)∥ = \frac{1}{2 sin(\frac{π}{4n+2})}  (1.29)

which means we can compute ∥Q_{n}(a)∥ for a = 0, \frac{1}{2}, 1. While the calculation of ∥M_{n}∥ and ∥Q_{n}(1)∥ is based on explicit formulae for all the eigenvalues and eigenvectors of certain associated operators, we could just pull them out of a hat. Instead, in Section 4, we discuss the motivation that led to our guess of eigenvectors, and in Section 5 explicitly prove Theorem 2.

Section 6 contains a number of remarks and extensions concerning Theorem 1, most importantly to numerical range concerns. Section 7 contains the application to random OPUC.

Acknowledgments. This work was done while B. Simon was a visitor at King’s College London. He would like to thank A. N. Pressley and E. B. Davies for the hospitality of King’s College, and the London Mathematical Society for partial support. The calculations of M. Stoiciu [19, 20] were an inspiration for our pursuing the estimate we found. We appreciate useful correspondence/discussions with M. Haase, N. Higham, R. Nagel, N. K. Nikolski, V. Totik, and L. N. Trefethen.

2. The Key Bound

Our goal in this section is to prove Theorem 4. A is an upper triangular n × n matrix. Let λ_{1}, …, λ_{n} be its diagonal elements. Since

det(z − A) = ∏_{j=1}^{n}(z − λ_{j})  (2.1)

the λ_{j}’s are the eigenvalues of A counting algebraic multiplicity. In particular,

sup_{j} |1 − λ_{j}|^{−1} = dist(1, spec(A))^{−1}  (2.2)

Define

C = (1 − A)^{−1} + (1 − A^{∗})^{−1} − 1  (2.3)

Proposition 2.1. Suppose ∥A∥ ≤ 1. Then

EIGENVALUE ESTIMATES 7

(a)

C_{jj} = |1 − λ_{j}|^{−2}(1 − |λ_{j}|^{2})
≤ 2|1 − λ_{j}|^{−1}  (2.4)

(b)

C ≥ 0

(c)

|C_{jk}| ≤ |C_{jj}|^{1/2}|C_{kk}|^{1/2}  (2.5)

(d) If j < k, then (1 − A)_{jk}^{−1} = C_{jk}.

Proof. (a) Since A is upper triangular,

[(1 − A)^{−1}]_{jj} = (1 − λ_{j})^{−1}  (2.6)

so (2.4) comes from

(1 − λ_{j})^{−1} + (1 − λ̄_{j})^{−1} − 1 = |1 − λ_{j}|^{−2}(1 − |λ_{j}|^{2})  (2.7)

and the fact that for |λ| ≤ 1,

|1 − λ|^{−1}(1 − |λ|^{2}) = (1 + |λ|)(1 − |λ|)(|1 − λ|^{−1})
≤ 2

since 1 − |λ| ≤ |1 − λ|.

(b) The operator analog of (2.7) is the direct computation

C = [(1 − A)^{−1}]^{∗}(1 − A^{∗}A)(1 − A)^{−1} ≥ 0  (2.8)

since ∥A∥ ≤ 1 implies A^{∗}A ≤ 1.

(c) This is true for any positive definite matrix.

(d) (1 − A^{∗})^{−1} is lower triangular and 1 is diagonal. □

Proof of Theorem 4. (1 − A)^{−1} is upper triangular so [(1 − A)^{−1}]_{kℓ} = 0 if k > ℓ. By (2.6) and (2.2),

|[(1 − A)^{−1}]_{kk}| = |1 − λ_{k}|^{−1} ≤ dist(1, spec(A))^{−1}  (2.9)

By (a), (c), (d) of the proposition, if k < ℓ,

|[(1 − A)^{−1}]_{kℓ}| ≤ [|1 − λ_{k}|^{−2}|1 − λ_{ℓ}|^{−2}(1 − |λ_{k}|^{2})(1 − |λ_{ℓ}|^{2})]^{1/2}
≤ 2[|1 − λ_{k}|^{−1}|1 − λ_{ℓ}|^{−1}]^{1/2}
≤ 2[dist(1, spec(A))]^{−1}

by (2.2). □

3. Upper Triangular Toeplitz Matrices

A Toeplitz matrix [1] is one that is constant along diagonals, that is, A_{jk} is a function of j − k. An n × n upper triangular Toeplitz matrix (UTTM) is thus of the form

(\begin{matrix}a_{0} & a_{1} & a_{2} & … & a_{n−1} \\ 0 & a_{0} & a_{1} & … & a_{n−2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & a_{0} \end{matrix})  (3.1)

These concern us because M_{n} is of this form and because the operators, A_{n}(a), of Theorem 3 will be of this form. In this section, after recalling the basics of UTTM, we will prove Theorem 3. Then we will state some results, essentially due to Schur [15], on the norms of UTTM that we will need in Section 5 in one calculation of the norm of M_{n}.

Given any function, f, which is analytic near zero, we write T_{n}(f) for the matrix in (3.1) if

f(z) = a_{0} + a_{1}z + ⋯ + a_{n−1}z^{n−1} + O(z^{n})  (3.2)

f is called a symbol for T_{n}(f).

We note that

T_{n}(fg) = T_{n}(f)T_{n}(g)  (3.3)

This can be seen by multiplying matrices and Taylor series or by manipulating projections on ℓ^{2} (see, e.g., Corollary 6.2.3 of [16]).

In addition, if f is analytic in {z | |z| < 1}, then

∥T_{n}(f)∥ ≤ sup_{|z|<1} |f(z)|  (3.4)

To see this, associate an analytic function

v(z) = v_{0} + v_{1}z + ⋯  (3.5)

to the vector φ_{n}(v) ∈ ℂ^{n} by

φ_{n}(v) = (v_{n−1}, v_{n−2}, …, v_{0})^{T}  (3.6)

and note that with ∥ · ∥_{2}, the H^{2} norm,

∥φ_{n}(v)∥ = inf{∥v∥_{2} | φ_{n} = φ_{n}(v)}  (3.7)
T_{n}(f)φ_{n}(v) = φ_{n}(fv)  (3.8)

and

∥fv∥_{2} ≤ ∥f∥_{∞}∥v∥_{2}  (3.9)

If N_{n} is given by (1.13), then T_{n}(f) = f(N_{n}), so an alternate proof of (3.4) may be based on von Neumann’s theorem; see Subsection 6E.

Proof of Theorem 3. For a with 0 < a < 1, define

f_{a}(z) = \frac{z + a}{1 + az}  (3.10)

and define

A_{n}(a) = T_{n}(f_{a})  (3.11)

Then f_{a}(e^{iθ}) = e^{iθ} \overline{(1 + ae^{iθ})}/(1 + ae^{iθ}) has |f_{a}(e^{iθ})| = 1, so sup_{|z|<1}|f_{a}(z)| = 1 and thus, by (3.4),

∥A_{n}(a)∥ ≤ 1  (3.12)

By (3.1),

spec(A_{n}(a)) = {f_{a}(0)} = {a}  (3.13)

By (3.5),

(1 − A_{n}(a))^{−1} = T_{n}((1 − f_{a}(z))^{−1})  (3.14)

Now

(1 − a)(1 − f_{a}(z))^{−1} = \frac{z + a}{1 − z}  (3.15)

so

lim_{a↑1} (1 − a)(1 − f_{a}(z))^{−1} = \frac{1 + z}{1 − z}  (3.16)

Thus,

lim_{a↑1} (1 − a)(1 − A_{n}(a))^{−1} = T_{n}(\frac{1 + z}{1 − z} ) = M_{n}  (3.17)

since (1 + z)/(1 − z) = 1 + 2z + 2z^{2} + · · ·. □

We now want to refine (3.4) to get equality for a suitable f. A key role is played by

Lemma 3.1. Let α ∈ 𝔻 and A an operator with \overline{α}^{−1} ∉ spec(A). Define

B = (A − α)(1 − \overline{α}A)^{−1}  (3.18)

Then

(1)  ∥B∥ ≤ 1 ⇔ ∥A∥ ≤ 1  (3.19)
(2)  ∥B∥ = 1 ⇔ ∥A∥ = 1  (3.20)

Proof. By a direct calculation,

1 − B^{∗}B = (1 − αA^{∗})^{−1}[(1 − |α|^{2})(1 − A^{∗}A)](1 − \overline{α}A)^{−1} (3.21)

(3.19) follows since 1 − B^{∗}B ≥ 0 ⇔ 1 − A^{∗}A ≥ 0, and (3.20) follows since (3.21) implies

inf_{∥φ∥=1} (φ, (1 − B^{∗}B)φ) = 0 ⇔ inf_{∥φ∥=1} (φ, (1 − A^{∗}A)φ) = 0  □

Remark. This lemma is further discussed in Subsection 6E.

Theorem 3.2. If A is an n×n UTTM with ∥A∥ ≤ 1, then there exists an analytic function, f, on 𝔻 such that

sup_{|z|<1} |f(z)| ≤ 1  (3.22)

and

A = T_{n}(f)  (3.23)

Proof. The proof is by induction on n. If n = 1, ∥A∥ ≤ 1 means |a_{0}| ≤ 1 and we can take f(z) ≡ a_{0}. For general n, ∥A∥ ≤ 1 means |a_{0}| ≤ 1. If |a_{0}| = 1, then A = a_{0}1 and we can take f(z) ≡ a_{0}. If a_{0} < 1, define B by (3.18) with α = a_{0}. B is a UTTM with zero diagonal terms, so

B = (\begin{matrix}0 & & & & B̃ \\ & . & . & . \\ 0 & & & & 0 \end{matrix})  (3.24)

where ∥B̃∥ = ∥B∥ ≤ 1 by the lemma.

By the induction hypothesis, B̃ = T_{n−1}(g) where

sup_{|z|<1} |g(z)| ≤ 1  (3.25)

Then (3.23) holds with

f = \frac{a_{0} + zg}{1 + \overline{a}_{0}zg}  (3.26)

(3.25) and (3.26) imply (3.22). □

Remarks. 1. By iterating f → g, we see that one constructs f via the Schur algorithm; see Section 1.3 of [16].

2. Combining this and (3.4), one obtains Schur’s celebrated result that a_{0} + a_{1}z + ⋯ + a_{n−1}z^{n−1} is the start of the Taylor series of a Schur function if and only if the matrix A of (3.1) obeys A^{∗}A ≤ 1. This result is intimately connected to Nehari’s theorem on the norm of Hankel operators [8, 13]; see Partington [12].

To state the last result of this section, we need a definition:

Definition. A Blaschke factor is a function on 𝔻 of the form

f(z, w) = \frac{z − w}{1 − \overline{w}z}  (3.27)

where w ∈ 𝔻. A (finite) Blaschke product is a function of the form

f(z) = ω ∏_{j=1}^{k} f(z, w_{k})  (3.28)

where ω ∈ ∂𝔻. k is called the order of f. We allow k = 0, in which case f(z) is a constant value in ∂𝔻.

Theorem 3.3. An n × n UTTM, A, has ∥A∥ = c if and only if A = T_{n}(f) for an f so that c^{−1}f is a Blaschke product of order k ≤ n − 1.

Proof. Without loss, we can take c = 1. The proof is by induction on n. If n = 1, k must be 0, and the theorem says |a_{0}| = 1 if and only if f(0) = ω ∈ ∂𝔻, which is true.

It is not hard to see that if f and f_{1} are related by

f_{1}(z) = z^{−1} \frac{f(z) − f(0)}{1 − \overline{f(0)} f(z)}

then f is a Blaschke product of order k ≥ 1 if and only if f_{1} is a Blaschke product of order k − 1.

Given A a UTTM with ∥A∥ ≤ 1, |a_{0}| = 1 if and only if A = T_{n}(a_{0}), that is, A is given by a Blaschke product of order 0. If |a_{0}| < 1, we define B by (3.18). ∥B∥ = 1 if and only if ∥A∥ = 1. B̃ given by (3.25) is related to A by A = T_{n}(f) if and only if B̃ = T_{n−1}(f_{1}). Thus, by induction, ∥A∥ = 1 if and only if f is a Blaschke product of order k ≤ n − 1. □

4. Inverse of Differential/Difference Operators

In this section and the next, we will find explicit formulae for the norms of M_{n} and Q_{n} ≡ Q_{n}(1) given by (1.28). Indeed, we will find all the eigenvalues and eigenvectors for |M_{n}| and |Q_{n}| where |A| = \sqrt{A^{∗}A}. A key to our finding this was understanding a kind of continuum limit of M_{n}: Let K be the Volterra-type operator on H = L^{2}([0, 1], dx) with integral kernel

K(x, y) = \begin{cases}1 & 0 ≤ x ≤ y ≤ 1 \\ 0 & 0 ≤ y < x < 1 \end{cases}

In some formal sense, K is a limit of either M_{n} or Q_{n}, but in a precise sense, M_{n} is a restriction of K:

Proposition 4.1. Let π_{n} be the projection of H onto the space of functions constant on each interval [\frac{j}{n}, \frac{j+1}{n}), j = 0, 1, …, n − 1. Then

π_{n}Kπ_{n}  (4.1)

is unitarily equivalent to \frac{1}{2}M_{n}/n. In particular,

∥M_{n}∥ ≤ 2n∥K∥  (4.2)
lim_{n→∞} \frac{∥M_{n}∥}{n} = 2∥K∥  (4.3)

Proof. Let {f_{j}^{(n)}}_{j=0}^{n−1} be the functions

f_{j}^{(n)}(x) = \begin{cases}\sqrt{n} & \frac{j}{n} ≤ x < \frac{j+1}{n} \\ 0 & otherwise \end{cases}  (4.4)

which form an orthonormal basis for Ran(π_{n}). Since

n⟨f_{j}^{(n)}, Kf_{k}^{(n)}⟩ = \frac{1}{2} (M_{n})_{jk}  (4.5)

we have the claimed unitary equivalence. (4.2) is immediate from ∥π_{n}Kπ_{n}∥ ≤ ∥K∥. (4.3) follows if we note s-lim_{n→∞} π_{n} = 1, so lim ∥π_{n}Kπ_{n}∥ = ∥K∥. □

Notice that

(Kf)(x) = ∫_{x}^{1} f(y) dy  (4.6)

so

\frac{d}{dx} (Kf) = f  Kf(1) = 0  (4.7)

and K is an inverse of a derivative. That means K^{∗}K will be the inverse of a second-order operator. Indeed,

(K^{∗}K)(x, y) = ∫_{0}^{1} \overline{K(z, x)} K(z, y) dz
= ∫_{0}^{min(x,y)} dz
= min(x, y)  (4.8)

which, as is well known, is the integral kernel of the inverse of −\frac{d^{2}}{dx^{2}} with u(0) = 0, u′(1) = 1 boundary conditions.

We can therefore write down a complete orthonormal basis of eigenfunctions for K^{∗}K:

φ_{n}(x) = sin(\frac{1}{2} (2n − 1)πx) n = 1, 2, …  (4.9)
(K^{∗}K)φ_{n} = \frac{4}{(2n − 1)^{2}π^{2}}  (4.10)

so

∥K∥ = ∥K^{∗}K∥^{1/2} = \frac{2}{π}  (4.11)

By (4.2), (4.3), we have

Corollary 4.2.

∥M_{n}∥ ≤ \frac{4n}{π}  (4.12)

lim_{n→∞} \frac{∥M_{n}∥}{n} = \frac{4}{π}  (4.13)

Of course, we will see this when we have proven Theorem 2, but it is interesting to have it now.

While M_{n} is related to differential operators via (4.5), we can compute the norm of Q_{n} by realizing it as the inverse of a difference operator. Specifically, let N_{n} be given by (1.13). Then

(1 − N_{n})^{−1} = 1 + N_{n} + N_{n}^{2} + ⋯ + N_{n}^{n−1} = Q_{n}  (4.14)

Theorem 4.3. Let

D_{n} = (1 − N_{n})(1 − N_{n})^{∗}  (4.15)

Then D_{n} has a complete set of eigenvectors:

v_{j}^{(ℓ)} = sin(\frac{π(2ℓ + 1)j}{2n + 1} )  j = 1, …, n; ℓ = 0, …, n − 1 (4.16)
D_{n}v^{(ℓ)} = 4 sin^{2}(\frac{π(2ℓ + 1)}{2(2n + 1)})v^{(ℓ)}  (4.17)
∥Q_{n}∥ = (min eigenvalue of D_{n})^{−1/2}
= [2 sin(\frac{π}{4n + 2})]^{−1}  (4.18)

Proof. By a direct calculation,

D_{n} = ( \begin{matrix}2 & −1 & 0 \\ −1 & 2 & −1 \\ 0 & −1 & 2 \\ & & & . & . & . \\ & & & & & & 2 & −1 & 0 \\ & & & & & & −1 & 2 & −1 \\ & & & & & & 0 & −1 & 1 \end{matrix} )  (4.19)

is a discrete Laplacian with Dirichlet boundary condition at 0 and Neumann at n. Since

− sin(q(j + 1)) + 2 sin(qj) − sin(q(j − 1)) = 4 sin^{2}(\frac{q}{2}) sin(qj)

(4.16)/(4.17) hold so long as q is such that sin(q(n+1)) = sin(qn), that is,

\frac{1}{2} [q(n + 1) + qn] = (ℓ + \frac{1}{2})π

or q = (2ℓ + 1)π/(2n + 1). □

Remark. For OPUC with dμ = dθ/2π, in the basis 1, z, …, z^{n−1}, A_{n} is given by the matrix, N_{n}, of (1.13), and so ∥(1−N_{n})^{−1}∥ = ∥Q_{n}∥ ∼ 2n/π. Thus, there are unit vectors, y_{n}, in this case with ∥(1−A_{n})y_{n}∥ ∼ π/2n.

5. The Norm of M_{n}

In this section, we will give two distinct but related proofs of Theorem 2. Both depend on a generating function relation:

Theorem 5.1. For θ ∈ (0, π) and z ∈ 𝔻, define

S_{θ}(z) = ∑_{j=0}^{∞} sin((2j + 1)θ)z^{j}  (5.1)

C_{θ}(z) = ∑_{j=0}^{∞} cos((2j + 1)θ)z^{j}  (5.2)

Then

\frac{1 + z}{1 − z} C_{θ}(z) = cot(θ)S_{θ}(z)  (5.3)

Proof. Let ω = e^{iθ} so, summing the geometric series,

S_{θ}(z) = (2i)^{−1} ∑_{j=0}^{∞}(ω^{2j+1}z^{j} − ω̄^{2j+1}z^{j})

= (2i)^{−1}[\frac{ω}{1 − zω^{2}} − \frac{ω̄}{1 − zω̄^{2}} ]  (5.4)
= \frac{sin(θ)(1 + z)}{(1 − zω^{2})(1 − zω̄^{2})}  (5.5)

For C_{ω}(z), the calculation is similar; in (5.4), (2i)^{−1} is replaced by (2)^{−1} and the minus sign becomes a plus:

C_{ω}(z) = \frac{cos(θ)(1 − z)}{(1 − zω^{2})(1 − zω̄^{2})}  (5.6)

(5.5) and (5.6) imply (5.3). □

Our first proof of Theorem 2 depends on looking at the Hankel matrix [12, 13]

M̃_{n} = (\begin{matrix}2 & 2 & … & 2 & 1 \\ 2 & 2 & … & 1 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 1 & 0 & … & 0 & 0 \end{matrix})  (5.7)

If W_{n} is the unitary permutation matrix

(W v)_{j} = v_{n+1−j}  (5.8)

then

M_{n} = M̃_{n}W M̃_{n} = M_{n}W (5.9)

and so

∥M_{n}∥ = ∥M̃_{n}∥  (5.10)

Here is our first proof of Theorem 2:

Theorem 5.2. Let

c_{j}^{(n;ℓ)} = cos((2ℓ + \frac{1}{2})\frac{π}{2n} (2j − 1)) j = 1, 2, …, n; ℓ = 0, …, n − 1

(5.11)

Then

M̃_{n}c^{(n;ℓ)} = cot((2ℓ + \frac{1}{2})\frac{π}{2n})c^{(n;ℓ)}  (5.12)

Thus,

∥M_{n}∥ = ∥M̃_{n}∥ = cot(\frac{π}{4n})  (5.13)

Proof. Let

c_{j}^{(n;θ)} = cos(θ(2j − 1)) j = 1, 2, …, n  (5.14)

and

s_{j}^{(n;θ)} = sin(θ(2j − 1))  j = 1, …, n  (5.15)

Then (5.3) implies that

M_{n}W c^{(n;θ)} = cot(θ)W s^{(n;θ)}  (5.16)

by looking at coefficients of 1, z, …, z^{n−1}. The W comes from (3.6)/(3.8). If

θ = \frac{π}{2} + 2ℓπ ℓ = 0, …, n − 1  (5.17)

then

W s^{(n;θ)} = c^{(n;θ)}  (5.18)

and (5.16) becomes (5.12).

Since M̃ is self-adjoint, (5.13) follows from (5.12) either by noting that max|cot((2ℓ + \frac{1}{2})\frac{π}{2n})| = cot(\frac{π}{4n}) or by noting that c^{(n;θ=π/4n)} is a positive eigenvector of a positive self-adjoint matrix, so its eigenvalue is the norm by the Perron-Frobenius theorem. □

Our second proof relies on the following known result (see Milovanić et al. [5], page 272, and references therein; this result is called the Eneström-Kakeya theorem):

Lemma 5.3. Suppose

0 < a_{0} < a_{1} < ⋯ < a_{n}  (5.19)

Then

P (z) = a_{0} + a_{1}z + ⋯ + a_{n}z^{n}  (5.20)

has all its zeros in 𝔻.

Theorem 5.4. Let

S^{(n)}(z) = ∑_{j=0}^{n−1} sin((2j + 1)\frac{π}{4n})z^{j}  (5.21)

C^{(n)}(z) = ∑_{j=0}^{n−1} cos((2j + 1)\frac{π}{4n})z^{j}  (5.22)

Then

b^{(n)}(z) = \frac{S^{(n)}(z)}{C^{(n)}(z)}  (5.23)

is a Blaschke product of order n − 1. Moreover,

cot(\frac{π}{4n})b^{n}(z) = 1 + 2 ∑_{j=1}^{n−1} z^{j} + O(z^{n})  (5.24)

and

∥M_{n}∥ = cot(\frac{π}{4n})  (5.25)

Proof. The coefficients of S^{(n)} obey (5.19) so, by the lemma, S^{(n)} has all its zeros in 𝔻. Moreover, by (5.18), C^{(n)}(z) = z^{n} \overline{S^{(n)}(1/z̄)}, which implies (5.23) is a Blaschke product.

(5.24) is just a translation of (5.3). (5.24) implies (5.25) by Theorem 3.3. □

6. Some Remarks and Extensions

In this section,we make some remarks that shed light on or extend Theorem 1, our main result.

A. An alternate proof. We give a simple proof of a weakened version of Theorem 4 but which suffices for applications like those in Section 7. This argument is related to ones in Section 3 of Nikolski [11].

Theorem 6.1. If ∥A∥ ≤ 1 and 1 ∉ spec(A), then

dist(1, spec(A))∥(1 − A)^{−1}∥ ≤ 2m  (6.1)

where m is the degree of the minimal polynomial for A.

Proof. We prove the result for ∥A∥ < 1. The general result follows by taking limits. We make repeated use of Lemma 3.1 which implies that if, for λ ∈ 𝔻, and we define

B(λ) = (\frac{A − λ}{1 − \overline{λ}A})(\frac{1 − \overline{λ}}{1 − λ})  (6.2)

then

∥B(λ)∥ ≤ 1  (6.3)

By algebra,

(1 − x)^{−1}[1 − \frac{x − λ}{1 − \overline{λ}x}(\frac{1 − \overline{λ}}{1 − λ})] = \frac{1}{1 − λ} [1 + \overline{λ}(\frac{x − λ}{1 − x\overline{λ}})] (6.4)

so, by Lemma 3.1 again,

∥(1 − A)^{−1}(1 − B(λ))∥ ≤ |1 − λ|^{−1}(1 + |λ|)  (6.5)

Now let ∏_{j=1}^{m}(x − λ_{j}) be the minimal polynomial for A. Then

∏_{j=1}^{m} B(λ_{j}) = 0

so

(1 − A)^{−1} = (1 − A)^{−1}[1 − ∏_{j=1}^{m} B_{j}(λ)]

= ∑_{j=1}^{m} (1 − A)^{−1}[1 − B_{j}(λ)] ∏_{k=j+1}^{m} B_{k}(λ)  (6.6)

(the empty product for j = m is interpreted as the identity operator) which, by (6.3) and (6.5), implies

LHS of (6.1) ≤ ∑_{j=1}^{m} dist(1, spec(A))|1 − λ_{j}|^{−1}(1 + |λ_{j}|)
≤ 2m

since 1+|λ_{j}| ≤ 2 and λ_{j} ∈ spec(A) so dist(1, spec(A))|1−λ_{j}|^{−1} ≤ 1. □

Remarks. 1. The factor (1 − \overline{λ})/(1 − λ) is taken in (6.2) so f_{λ}(z) = (z − λ)(1 − \overline{λ}z)^{−1}(1 − \overline{λ})(1 − λ)^{−1} has 1 − f_{λ}(1) = 0.

2. In place of the algebra (6.4), one can compute that the sup_{|z|<1} LHS of (6.4) is |1 − λ|^{−1}[1 + |λ|] and use von Neumann’s theorem as discussed in Subsection E below.

B. Minimal polynomials. While the constant 2 in (6.1) is worse than 4/π in (1.19)/(1.21), (6.1) appears to be stronger in that m, not n, appears, but we can also strengthen (1.19) in this way:

Theorem 6.2. If ∥A∥ ≤ 1, 1 ∉ spec(A), and m is the degree of the minimal polynomial for A, then

dist(1, spec(A))∥(1 − A)^{−1}∥ ≤ cot(\frac{π}{4m})  (6.7)

Proof. Let ∥y∥ = 1. Since A^{m}y is a linear combination of {A^{j}y}_{j=0}^{m−1}, the cyclic subspace, V_{y}, has dim(V_{y}) ≡ m_{y} ≤ m. Since A ↾ V_{y} is an operator of a space of dimension m_{y}, we have

dist(1, spec(A))∥(1 − A)^{−1}y∥ ≤ c(m_{y}) = cot(\frac{π}{4m_{y}} )
≤ cot(\frac{π}{4m})  □

C. Numerical range. For any bounded operator, A, on a Hilbert space, the numerical range, Num(A), is defined by

Num(A) = {⟨φ, Aφ⟩ | ∥φ∥ = 1}  (6.8)

It is a bounded convex set (see [3, p. 150]), and when A is a finite matrix, also closed. Theorem 1 can be improved to read:

Theorem 6.3. Let M̃_{n} be the set of pairs (A, z) where A is an n × n matrix, z ∈ ℂ with

z ∉ spec(A)  z ∉ Num(A)^{int}  (6.9)

Then

sup ; M̃_{n} dist(z, spec(A))∥(A − z)^{−1}∥ = cot(\frac{π}{4n})  (6.10)

Remarks. 1. Since Num(A) ⊂ {z | |z| ≤ ∥A∥}, M_{n} ⊂ M̃_{n}, and this is a strict improvement of (1.19).

2. We need only prove

dist(z, spec(A))∥(A − z)^{−1}∥ ≤ cot(\frac{π}{4n})

since the equality then follows from M_{n} ⊂ M̃_{n}.

3. By replacing A by e^{iθ}(A − z) for suitable θ and z, we need only prove

Re(A) ≥ 0, 0 ∉ spec(A) ⇒ dist(0, spec(A))∥A^{−1}∥ ≤ cot(\frac{π}{4n}) (6.11)

for by convexity of Num(A), if z ∉ Num(A)^{int}, there is a half-plane, P, with Num(A) ⊂ P and z ∈ ∂P. It is (6.11) we will prove below.

First Proof of Theorem 6.3. Let

C = A^{−1} + (A^{∗})^{−1}  (6.12)
= (A^{∗})^{−1}2 Re(A)(A)^{−1} ≥ 0  (6.13)

Thus,

|C_{jk}| ≤ |C_{jj}|^{1/2}|C_{kk}|^{1/2}  (6.14)

Now just follow the proof of Theorem 4 in Section 2. □

Second Proof of Theorem 6.3. We use Cayley transforms. For 0 < s, define

B(s) = (1 − sA)(1 + sA)^{−1}  (6.15)

Since

∥(1 + sA)φ∥^{2} − ∥(1 − sA)φ∥^{2} = 4s Re(φ, Aφ) ≥ 0

we have that

∥B(s)∥ ≤ 1  (6.16)

Because

1 − B(s) = 2sA(1 + sA)^{−1}  (6.17)

we have for s small that

dist(1, spec(B(s))) = 2s dist(0, spec(A)) + O(s^{2})  (6.18)

Thus, by Theorem 1,

2s dist(0, spec(A))∥(1 − B(s))^{−1}∥ ≤ cot(\frac{π}{4n}) + O(s)  (6.19)

By (6.17),

(1 − B(s))^{−1} = (2s)^{−1}[A^{−1} + s]

so

∥A^{−1}∥ ≤ |s| + 2s∥(1 − B(s))^{−1}∥  (6.20)

This plus (6.18) implies (6.11) as s ↓ 0. □

D. Bounded powers. We note that there is also a result if

sup_{m≥0} ∥A^{m}∥ = c < ∞  (6.21)

We suspect the 3/2 power in the following is not optimal. We note that one can also use this method if ∥A^{m}∥ is polynomially bounded in m.

Theorem 6.4. If (6.21) holds, then

∥(1 − A)^{−1}∥ ≤ c(3n)^{3/2}dist(1, spec(A))^{−3/2}  (6.22)

Proof. By the argument of Section 1 (using (1.11)), this is equivalent to

dist(1, spec(A)) ≤ 3n(c∥(1 − A)y∥)^{2/3}  (6.23)

for all unit vectors y.

Define for 1 < r,

⟨f, g⟩_{r} = ∑_{m=0}^{∞} r^{−2m}⟨A^{m}f, A^{m}g⟩  (6.24)

By (6.21),

∥f∥ ≤ ∥f∥_{r} ≤ cr(r^{2} − 1)^{−1/2}∥f∥  (6.25)

By (6.24),

∥Af∥_{r}^{2} ≤ r^{2}∥f∥_{r}^{2}  (6.26)

so

∥A∥_{r} ≤ r  (6.27)

so if C = r^{−1}A, then

∥C∥_{r} ≤ 1  (6.28)

Clearly, for ∥y∥ = 1 ≤ ∥y∥_{r},

∥Cy − y∥_{r} ≤ |r^{−1} − 1| ∥y∥_{r} + r^{−1}∥(A − 1)y∥_{r}
≤ |r^{−1} − 1| ∥y∥_{r} + c(r^{2} − 1)^{−1/2}∥(A − 1)y∥
≤ ((r − 1) + c[2(r − 1)]^{−1/2}∥(A − 1)y∥)∥y∥_{r}  (6.29)

It follows by Theorem 1 and the fact that spec(A) is independent of ∥ · ∥_{r} that

dist(1, r^{−1}spec(A)) ≤ \frac{4n}{π} \{c∥(A − 1)y∥(2(r − 1))^{−1/2} + (r − 1)\} (6.30)

and thus

dist(1, spec(A)) ≤ (r − 1) + \frac{4π}{n} \{c∥(A − 1)y∥(2(r − 1))^{−1/2} + (r − 1)\} (6.31)

Choosing r = 1+ \frac{1}{2}(c∥(A−1)y∥)^{2/3} and using \frac{1}{2} + \frac{6n}{π} ≤ 3n, we obtain (6.23). □

E. Von Neumann’s theorem. Lemma 3.1 is a special case of a theorem of von Neumann. The now standard proof of this result uses Nagy dilations [22]; we have found a simple alternative that relies on

Lemma 6.5. For any A, with ∥A∥ < 1 and A = U|A|, and U unitary, there exists an operator-valued function, g, analytic in a neighborhood of \overline{𝔻} so that g(e^{iθ}) is unitary and g(0) = A.

Proof. Let

g(z) = U[\frac{z + |A|}{1 + z|A|}]  (6.32)

The factor in […] is unitary if z = e^{iθ}, since

(e^{iθ} + |A|)^{∗}(e^{iθ} + |A|) = 1 + A^{∗}A + 2 cos θ|A|
= (1 + e^{iθ}|A|)^{∗}(1 + e^{iθ}|A|)  □

Theorem 6.6 (von Neumann [24]). Let f: 𝔻 → 𝔻. If ∥A∥ < 1, define f(A) by

f(z) = ∑_{n=0}^{∞} a_{n}z^{n}  f(A) ≡ ∑_{n=0}^{∞} a_{n}A^{n}  (6.33)

Then

∥f(A)∥ ≤ 1  (6.34)

Proof of von Neumann’s theorem, given the lemma. Suppose first that A obeys the hypotheses of the lemma. By a limiting argument, suppose f is analytic in a neighborhood of \overline{𝔻}. Applying the maximum principle to f(g(z)), we see

∥f(A)∥ = ∥f(g(0))∥ ≤ sup_{θ} ∥f(g(e^{iθ}))∥
= sup_{θ} |f(e^{iθ})| ≤ 1  (6.35)

where (6.35) uses the spectral theorem for the unitary g(e^{iθ}).

For general A, if Ã = A⊕0 on H⊕H, then Ã = U|Ã| with U unitary and we obtain ∥f(Ã)∥ ≤ 1. But f(Ã) = f(A) ⊕ 0. □

Remarks. 1. In general, A = V |A| with V a partial isometry. We can extend this to a unitary U so long as dim(Ran(V)^{⊥}) = dim(ker(V)^{⊥}). This is automatic in the finite-dimensional case and also if dim(H) = ∞ for A ⊕ 0 since then both spaces are infinite-dimensional.

2. This proof is close to one of Nelson [9] who also uses the maximum principle and polar decomposition, but uses a different method for interpolating the self-adjoint part (see also Nikolski [10]).

7. Zeros of Random OPUC

In this section, we apply Theorem 1 to obtain results on certain OPUC. We begin by recalling the recursion relations for OPUC [16, 17, 18]. For each non-trivial probability measure, dμ, on ∂𝔻, there is a sequence of complex numbers, {α_{n}(dμ)}_{n=0}^{∞}, called Verblunsky coefficients so that

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (7.1)

where

Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}  (7.2)

The α_{n} obey |α_{n}| < 1 and Verblunsky’s theorem [16, 18] says that μ ↦ {α_{n}(dμ)}_{n=0}^{∞} is a bicontinuous bijection from the non-trivial measures on ∂𝔻 with the topology of vague convergence to 𝔻^{∞} with the product topology.

For each ρ in (0, 1), we define the ρ-model to be the set of random Verblunsky coefficients where α_{n} are independent, identically distributed random variables, each uniformly distributed in {z | |z| ≤ ρ}. A point in the model space of α’s will be denoted ω; Φ_{n}(z; ω) will be the corresponding OPUC and {z_{j}^{(n)}(ω)}_{j=1}^{n} the zeros of Φ_{n} counting multiplicity. Our results here depend heavily on earlier results of Stoiciu [19, 20], who studied a closely related problem (see below). In turn, Stoiciu relied, in part, on earlier work on zeros of random Schrödinger operators [7, 6].

We will prove the following three theorems:

Theorem 7.1. Let 0 < ρ < 1. Let k ∈ {1, 2, … }. Then for a.e. ω in the ρ-model,

lim sup_{n→∞} \frac{#\{j | |z_{j}^{(n)}(ω)| < 1 − n^{−k}\}}{[log(n)]^{2}} < ∞  (7.3)

Thus, the overwhelming bulk of zeros are polynomially close to ∂𝔻. If we look at a small slice of argument, we can say more:

Theorem 7.2. Let 0 < ρ < 1. Let θ_{0} ∈ [0, 2π) and a < b real. Let η < 1. Then with probability 1, for large n, there are no zeros in \{z | arg z ∈ (θ_{0} + \frac{2πa}{n}, θ_{0} + \frac{2πb}{n}); |z| < 1 − exp(−n^{η})\}.

Finally and most importantly, we can describe the statistical distribution of the arguments:

Theorem 7.3. Let 0 < ρ < 1. Let θ_{0} ∈ [0, 2π). Let a_{1} < b_{1} ≤ a_{2} < b_{2} ≤ ⋯ ≤ a_{ℓ} < b_{ℓ} and let k_{1}, …, k_{ℓ} be in {0, 1, 2, … }. Then as n → ∞,

Prob(#(j | arg z_{j}^{(n)}(ω) ∈ (θ_{0}+\frac{2πa_{m}}{n} , θ_{0}+\frac{2πb_{n}}{n} )) = k_{m} for m = 1, …, ℓ

(7.4)

converges to

∏_{m=1}^{ℓ} \frac{(b_{m} − a_{m})^{k_{m}}}{k_{m}!} e^{−(b_{m}−a_{m})}  (7.5)

This says the zeros are asymptotically Poisson distributed. As we stated, our proofs rely on ideas of Stoiciu, essentially using Theorem 1 to complete his program. To state the results of his that we use, we need a definition.

For β ∈ ∂𝔻, the paraorthogonal polynomials (POPUC) are defined by

Φ_{n}^{(β)}(z) = Φ_{n−1}(z) − β̄Φ_{n−1}^{∗}(z)  (7.6)

These have zeros on ∂𝔻. Indeed, they are eigenvalues of a rank one unitary perturbation of the operator A_{n} of (1.6). We extend the ρmodel to include an additional set of independent parameters {β_{j}}_{j=0}^{∞} in ∂𝔻, each uniformly distributed on ∂𝔻. z̃_{j}^{(n)}(ω) denotes the zeros of Φ_{n}^{(β_{n})}(z; ω). Stoiciu [19, 20] completely analyzed these POPUC zeros. We will need three of his results:

Theorem 7.4 (= Theorem 6.1.3 of [20] = Theorem 6.3 of [19]). Let I be an interval in ∂𝔻. Then

Prob(2 or more z̃_{j}^{(n)}(ω) lie in I) ≤ \frac{1}{2}(\frac{n|I|}{2π})^{2}  (7.7)

where |I| is the dθ measure of I.

For the next theorem, we need the fact that there is an explicit realization of A_{n} and the associated rank one perturbations as n × n complex CMV matrices (see [2, 16, 17, 18]), C_{n}, whose eigenvalues are the z_{j}^{n}, and C̃_{n}^{(β_{n})} whose eigenvalues are the z̃_{j}^{n}, so that

∥(C_{n} − C_{n}^{(β_{n})})φ∥ ≤ |φ_{n−1}| + |φ_{n}|  (7.8)

The next theorem uses the components so (7.8) holds.

Theorem 7.5 (= Theorem 1.1.2 of [20] = Theorem 2.2 of [19]). There exists a constant D_{2} (depending only on ρ) so that for every eigenvector φ^{(j,ω;n)} of C̃_{n}^{(β_{n})}, we have for

|m − m(φ^{(j,ω;n)})| ≥ D_{2}(log n)  (7.9)

that

|φ_{m}^{(j,ω;n)}| ≤ C_{ω}e^{−4|m−m(φ^{(j,ω;n)})|/D_{2}}  (7.10)

where C_{ω} is an a.e. finite constant and

m(φ) = first k so |φ_{k}| = max_{m} |φ_{m}|  (7.11)

We will also need the results that Stoiciu proves along the way that for each C_{0},

{ω | C_{ω} < C_{0}} ≡ Ω_{C_{0}}  (7.12)

is invariant under rotation of the measures dμ_{ω}, and that for each C_{0} fixed and all ω ∈ Ω_{C_{0}},

#(j | m(φ^{(j,ω;n)}) = m_{0}) ≤ D_{3}(log n)  (7.13)

where D_{3} is only C_{0}-dependent and is independent of ω, m_{0}, and n. (7.13) comes from the fact that, by (7.10), for D_{3} only depending on C_{0},

∑_{|m−m(φ)|≥ \frac{1}{4} D_{3}(log n)} |φ_{m}|^{2} ≤ \frac{1}{2}  (7.14)

so, by (7.11), for φ’s with m(φ) = m_{0},

\frac{1}{2} D_{3}(log n)|φ_{m_{0}}|^{2} ≥ \frac{1}{2}  (7.15)

which, given

∑_{φ} |φ_{m_{0}}|^{2} = 1  (7.16)

implies (7.13).

The last of Stoiciu’s results we will need is

Theorem 7.6 (= Theorem 1.0.6 of [20] = Theorem 1.1 of [19]). For θ_{0} ∈ [0, 2π) and a_{1} < b_{1} ≤ a_{2} < b_{2} ≤ ⋯ ≤ a_{ℓ} < b_{ℓ} and k_{1}, …, k_{ℓ} in {0, 1, 2, … }, we have, as n → ∞, that (7.4) with z_{j}^{(n)} replaced by z̃_{j}^{(n)} converges to (7.5).

With this background out of the way, we begin the proofs of the new Theorems 7.1–7.3 with

Theorem 7.7. Fix ρ ∈ (0, 1). Then for a.e. ω, there exists N_{ω} so if n ≥ N_{ω}, then

min_{j≠k} |z̃_{j}^{(n)} − z̃_{k}^{(n)}| ≥ 2n^{−4}  (7.17)

Remark. n^{−3−ε} will work in place of n^{−4}.

Proof. For each n, cover ∂𝔻 by two sets of intervals of size 4n^{−4}: one set non-overlapping, except at the end, starting with [0, 4n^{−4}] and the other set starting with [2n^{−4}, 6n^{−4}]. If (7.17) fails for some n, then there are two zeros within one of these intervals. By (7.7), the probability of two zeros in one of these intervals is O((nn^{−4})^{2}) = O(n^{−6}). The number of intervals at order n is O(n^{4}). Since ∑_{n=1}^{∞} n^{4}n^{−6} < ∞, the sum of the probabilities of two zeros in an interval is summable. By the Borel-Cantelli lemma [21] for a.e. ω, only finitely many intervals have two zeros. Hence, for large n, (7.17) holds. □

Proof of Theorem 7.1. Obviously, if (7.3) holds for some k, it holds for all smaller k, so we will prove it for k ≥ 4. We also need only prove it on any Ω_{C_{0}} given by (7.12) since ∪Ω_{C_{0}} has probability 1 by Theorem 7.5. Consider those φ^{(j,ω;n)} with

|m(φ^{(j,ω;n)}) − n| ≥ K(log n)  (7.18)

By (7.13), the number of j for which (7.18) fails is O((log n)^{2}). By (7.10) and (7.8) and the fact that φ is a unit eigenfunction, then

∥(C_{n} − z̃_{j}^{(n)})φ^{(j,ω;n)}∥ ≤ 2C_{ω}n^{−4K/D_{2}}  (7.19)

so picking K large enough and n large enough that \frac{4}{π}2C_{ω}n^{−1} < 1, we have

∥(C_{n} − z̃_{j}^{(n)})φ^{(j,ω;n)}∥ ≤ \frac{π}{4n} n^{−k}  (7.20)

Thus, by Theorem 1 and ∥C_{n}∥ = 1 = |z̃_{j}^{(n)}|, we see that for each j obeying (7.18), there is a z_{j}^{(n)} so

|z_{j}^{(n)} − z̃_{j}^{(n)}| ≤ n^{−k}  (7.21)

By Theorem 7.7 and k ≥ 4, the z_{j}^{(n)} are distinct for n large, so we have n − O((log n)^{2}) zeros with |z_{j}^{(n)}| ≥ 1 − n^{−k}. This is (7.3). □

Proof of Theorem 7.2. In place of (7.18), we look for φ’s so

|m(φ^{(j,ω;n)}) − n| ≥ \frac{D_{2}}{2} n^{1−η}  (7.22)

For such j’s, using the above arguments, there are zeros z_{j}^{(n)} with

|z_{j}^{(n)} − z̃_{j}^{(n)}| ≤ C_{ω} exp(−2n^{η})  (7.23)

□

As in Stoiciu [19, 20], the distribution of z̃_{j}^{(n)} for which (7.22) fails is rotation invariant. Since the number is O(n^{1−η} log n) out of O(n) zeros, the probability of any of these had zeros lying in {z | arg z ∈ (θ_{0} + \frac{2πa}{n}, θ_{0} + \frac{2πb}{n})\} goes to zero as n → ∞.

Proof of Theorem 7.3. By the last proof, the zeros of Φ_{n} with the given arguments lie within O(e^{−n^{η}}) of those of Φ_{n}^{(β)} and, by Theorem 7.7, these zeros are distinct. Theorem 7.6 completes the proof if one gets upper and lower bounds by slightly increasing/decreasing the intervals on an O(1/n) scale. □

We close with the remark about improving these theorems. While (7.13) is the best one can hope for as a uniform bound, with overwhelming probability the number should be bounded. Thus, we expect in Theorem 7.1 that one can obtain O((log n)^{−1}) in place of O((log n)^{−2}). It is possible in Theorem 7.2 that one can improve O(e^{−n^{η}}) for all η ∈ 1 to O(e^{−An}) for some A.

References

[1] A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer, Berlin, 1990.

[2] M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29–56.

[3] E. B. Davies, One-Parameter Semigroups, London Mathematical Society Monographs, 15, Academic Press, London-New York, 1980.

[4] P. Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math. 4 (1962), 24–40.

[5] G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publishing, River Edge, NJ, 1994.

[6] N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), 709–725.

[7] S. A. Molchanov, The local structure of the spectrum of the one-dimensional Schrödinger operator, Comm. Math. Phys. 78 (1980/81), 429–446.

[8] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153–162.

[9] E. Nelson, The distinguished boundary of the unit operator ball, Proc. Amer. Math. Soc. 12 (1961), 994–995.

[10] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Vol. 2: Model Operators and Systems, Mathematical Surveys and Monographs, 93, American Mathematical Society, Providence, RI, 2002.

[11] N. K. Nikolski, Condition numbers of large matrices, and analytic capacities, to appear in St. Petersburg Math. J.

[12] J. R. Partington, An Introduction to Hankel Operators, London Mathematical Society Student Texts, 13, Cambridge University Press, Cambridge, 1988.

[13] V. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Math., Springer, New York, 2003.

[14] Pseudospectra Gateway,

http://web.comlab.ox.ac.uk/projects/pseudospectra/

[15] I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I, II, J. Reine Angew. Math. 147 (1917), 205–232; 148 (1918), 122– 145. English translation in “I. Schur Methods in Operator Theory and Signal Processing” (edited by I. Gohberg), pp. 31–59, 66–88, Operator Theory: Advances and Applications, 18, Birkhäuser, Basel, 1986.

[16] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.

[17] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.

[18] B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42 (2005), 431–460.

[19] M. Stoiciu, The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle, to appear in J. Approx. Theory.

[20] M. Stoiciu, Zeros of Random Orthogonal Polynomials on the Unit Circle, Ph.D. dissertation, 2005.

http://etd.caltech.edu/etd/available/etd-05272005-110242/

[21] D. Stroock, A Concise Introduction to the Theory of Integration, Series in Pure Math., 12, World Scientific Publishing, River Edge, NJ, 1990.

[22] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing, Amsterdam-London; American Elsevier Publishing, New York; Akadémiai Kiadó, Budapest, 1970.

[23] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Non-normal Matrices and Operators, Princeton University Press, Princeton, NJ, expected 2005.

[24] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281.