## 1 Introduction

### 1.1 Motivation

One of the most important results in operator theory is due to John von Neumann and states that for a fixed contraction on a Hilbert space, the operator norm satisfies

where is the complex unit disk and denotes the space of all one-variable polynomials with complex coefficients.

Variations of von Neumann’s inequality can be extremely useful and thus are frequently the object of study. Matsaev’s conjecture (see [31]), for example, asserts that for every contraction on (where is a measure space and ) and the unilateral shift operator defined by , the following inequality holds:

For and , it is not difficult to see that this is true, and for it is equivalent to von Neumann’s inequality. However, Drury [13] showed that Matsaev’s conjecture fails for .

Von Neumann’s inequality can be reformulated for a general bounded operator as

One may ask whether the supremum can instead be taken over subsets of , such as the spectrum , by possibly allowing for an absolute multiplicative constant in the inequality. By Crouzeix’s theorem [9], we may choose the subset to be the numerical range of , but it is still an open question as to what the best multiplicative constant is. This problem is known as Crouzeix’s conjecture.
To state it precisely, let
be an matrix with complex entries. Let denote its *numerical range*
and its numerical radius,

where refers to the Euclidean inner product and to its induced norm.
Then always contains the *spectrum* of , denoted by .
However, often encodes significantly more information about .
For example, is Hermitian if and only . Further,
if is normal, then is the convex hull of the eigenvalues of .
Thus, it seems plausible that the value of on
could be used to control .^{1}^{1}1From now on
will denote the 2-norm for vectors and the corresponding operator norm for
matrices: , which is also the largest
singular value of .
Indeed, in [8, 9], Crouzeix showed that

(1) |

with and he *conjectured* that the *best constant* in
(1) is .
Recently, Crouzeix and Palencia [10] proved that
(1) holds with , which is the best general constant
known so far.
Since every *normal matrix* is unitarily equivalent to a diagonal matrix,
it is clear that
(1) holds with in this case
and the supremum can be taken over the eigenvalues only.
It should be noted that if Crouzeix’s conjecture holds for matrices,
then it automatically holds for bounded operators on every Hilbert space [9].
Also, if it holds for all polynomials, then it holds for all
functions analytic in the interior of and continuous on the boundary,
since such functions can be arbitrarily well approximated by polynomials
[28, 29].

Crouzeix’s conjecture has inspired a great deal of mathematics and there are now several classes of matrices for which Crouzeix’s conjecture has been proved (see, for example, [2, 4, 5, 6, 8, 12, 25]). In particular, the conjecture is true for matrices as well as matrices of the form or where is a complex number, is a diagonal matrix, and is a permutation matrix (see [5], [8], and [23]), tridiagonal Toeplitz matrices and matrices in this class with some diagonal entries taken equal to zero, [23].

It is easy to show that the conjecture holds for Jordan blocks with zeros along the diagonal; it was later shown in [6] that the conjecture also holds for perturbed Jordan blocks:

These two classes of matrices (Jordan blocks and perturbed Jordan blocks when
) are special cases of operators known as *compressions of
the shift operator*. To define a general compressed shift, let denote the
usual Hardy space
of holomorphic functions on
and the shift operator defined
on
by . Beurling showed that the (closed) nontrivial invariant
subspaces for are of the form , where is an inner
function. Therefore, the invariant subspaces of the adjoint, , are of
the form , where denotes the
orthogonal complement. The associated compressed shift operator is defined by If
is a finite Blaschke product with , that is, if

then is finite-dimensional and can be represented as an matrix. Because much is known about the numerical ranges of these ([11, 20, 21, 30]), it is natural to consider Crouzeix’s conjecture for them. Moreover, Sz.-Nagy and Foias [37] showed that every completely non-unitary contraction of class with defect is unitarily equivalent to some . Thus, establishing Crouzeix’s conjecture for such would imply it for a large collection of matrices at once.

In their paper [10], Crouzeix and Palencia used clever complex analysis techniques to study (1). Specifically, for an open, convex set with smooth boundary containing , they showed that for all holomorphic on and continuous up to the boundary,

which implies (1) with . An maximizing among all holomorphic functions on with is called extremal for the pair . By a normal-families argument, such an extremal always exists. Recall that for a complex function defined and continuous on the boundary of a set , the Cauchy transform of on is defined by

A key part of the Crouzeix-Palencia proof showed that if and , then

where the asterisk denotes the adjoint (or conjugate transpose) of an operator. It follows that if it were true that for extremal , we have

(2) |

then Crouzeix’s conjecture would follow for . This motivates our study of such extremal below. See also [33] for an analysis of the Crouzeix-Palencia proof.

### 1.2 Main Results

In this paper, we provide a survey of our recent investigations related to Crouzeix’s conjecture; in particular, we derive specific bounds for , where is chosen in an appropriate algebra, as well as properties of related extremal functions and associated vectors. Throughout, we will use compressions of the shift to both motivate and illustrate our results. While these investigations have yielded a number of results, many questions remain open. Below and throughout this paper, we will highlight these open questions and invite any interested parties to take up their study.

Recall that if has distinct eigenvalues, then factors as for some diagonal and . In Section 2, we use this formula paired with classical results about function theory on to study . First, in Subsection 2.1, we let be a contraction with eigenvalues that are pseudohyperbolically well separated (see (6) below) and let denote a constant depending on the separation of the eigenvalues given in (7). Then in Theorem 2.2

, we combine results from interpolation theory with von Neumann’s inequality to deduce the existence of a constant

such that(3) |

where as . For sufficiently close
to , this implies that the matrix is *near normal*

in the sense that it has a well-conditioned matrix of eigenvectors; that is,

is of moderate size. We thus have a criterion for near normality in terms of the eigenvalues and the largest singular value (i.e., the operator norm) of the matrix. Clearly, if , then Crouzeix’s conjecture holds for since

(4) |

For a matrix with distinct eigenvalues, one can similarly define the minimal condition number of an eigenvector matrix of by

(5) |

and as in (4), if , then Crouzeix’s conjecture holds for . In Subsection 2.2, we study this setup for matrices that are representations of the compression of the shift , associated with a finite Blaschke product with distinct zeros in . First, in Theorem 2.4, we provide tractable formulas for specific and . Then, in Theorem 2.6, we use these formulas to obtain a bound on in terms of the separation of the zeros of . Because compressions of the shift are quite important, we pose the following question:

###### Question 1.1.

What is the minimal condition number of an eigenvector matrix of for a general finite Blaschke product with distinct zeros?

In Section 3, we turn to extremal functions and vectors. Let
be a bounded simply connected domain with smooth boundary containing
the spectrum, , of . We are interested in studying
, where the supremum is taken over all ,
the closed
unit
ball of bounded holomorphic functions on As before, an for which
the supremum is attained is called *extremal* for the pair
and any non-zero vector where
(i.e., any right singular vector of associated with the
largest singular value) is called an associated *extremal vector*.
Such vectors are also called *maximal*, see for example [34].
Crouzeix [8, Theorem 2.1] showed that an extremal
function
for is necessarily of the form
, where is a bijective conformal map of onto the
open unit disk , and is a Blaschke product of degree at most
.

In Section 3.1, we consider compressions of the shift for a finite Blaschke product and . Theorem 3.1, which is proved in [18], characterizes the extremal functions for ; here, we provide a simple proof in the case where has distinct zeros. Meanwhile, Theorem 3.2 characterizes the associated extremal vectors. In Section 3.2, we consider a general matrix and set , assuming that . Characterizing the extremal functions in this situation is significantly more complicated. Instead of tackling that problem in its entirety, we investigate the possible degrees of an extremal Blaschke product . In Section 3.2, we give an example of a matrix (defined in (12)) for which the only extremal functions have (maximal) degree , and in Theorem 3.8 we show that there is an open set of matrices for which the extremal Blaschke products have maximal degree. The following question is still open:

###### Question 1.2.

Given an matrix with and setting , what are the degree(s) of the associated extremal Blaschke product(s) ?

In Section 4, we return to a general and and study the associated extremal functions and vectors. It is already known that extremal functions enjoy the following orthogonality property, see [4, Theorem 5.1]: if is extremal for , if , and if is a unit vector on which attains its norm (i.e. an extremal unit vector), then . We generalize this result in Section 4.1. In particular, if is an extremal unit vector, Theorem 4.1 shows that if is a factorization of with each , then

This can be viewed as a sort of cancellation theorem, particularly in the case when . In Theorem 4.4, we prove a similar result for functions extremal with respect to the numerical radius.

In Section 4.2, we provide representation theorems for extremal vectors. For example, using an extremal function for , we obtain Theorem 4.5, which shows that for each associated extremal unit vector

, there is a unique Borel probability measure

defined on such that for all , we haveA similar result holds for vectors that are extremal with respect to the numerical radius. To demonstrate Theorems 4.1 and 4.5, we apply them to the extremal functions and vectors for , see Examples 4.3 and 4.8. Furthermore, the connections between Crouzeix’s conjecture and the structure of extremal functions and vectors mentioned earlier motivate the following question:

###### Question 1.3.

More open questions related to these topics are delineated throughout the rest of the paper.

## 2 Crouzeix’s Conjecture via Pointwise Bounds and Condition Numbers

In this section, is a contraction with distinct eigenvalues in . Note that factors as with diagonal and, for any defined on the eigenvalues of , we have . To measure how separated the eigenvalues are, we define the *pseudohyperbolic distance* between and in as

(6) |

We use classical function theory to study for certain classes of functions . Recall that is the usual Hardy space on a domain , and let the algebra consisting of bounded analytic functions on be denoted by , with closed unit ball . When , we often simply write and .

### 2.1 Bounds via Interpolation Theory

For a finite or infinite sequence of points in , we let

(7) |

where is called the *separation constant* corresponding to . The following result due to J. P. Earl connects this separation constant to interpolation problems:

###### Theorem 2.1 ([15]).

Let be a sequence in with separation constant and let be a bounded sequence of complex numbers. Then there exists solving the interpolation problem , for , with , where

Theorem 2.1 paired with von Neumann’s inequality can be used to provide a bound on . In what follows, for an matrix with distinct eigenvalues in , we define .

###### Theorem 2.2.

Let be an matrix with with distinct eigenvalues and suppose that . If , then

By rescaling, there is a version of Theorem 2.2 for general matrices with distinct eigenvalues and . Note however, that is not invariant under mappings for .

###### Proof.

Write . By Theorem 2.1 applied to , there exists such that for all and . Since on and the eigenvalues of are distinct, we have . And, since is a contraction, von Neumann’s inequality yields . Putting this together we have

the desired bound. ∎

In Theorem 2.1, the is a decreasing function of that tends to as . Thus, in Theorem 2.2, when the eigenvalues of are far apart, pseudohyperbolically speaking, the constant is close to and we need only consider the behavior of on the spectrum of

to get an estimate on

. Moreover, a computation shows that for , we have . Thus, for matrices with well-separated eigenvalues, the following strong form of Crouzeix’s conjecture holds.###### Corollary 2.3.

Let be an matrix with , , and . Then for , we have

Note, however, that the assumption implies that the pseudohyperbolic distance between each pair of eigenvalues is at least

. If the eigenvalues are uniformly distributed around a circle of radius

about the origin, for example, then when , this means that must be greater than about ; for , ; for , ; for , ; for , , etc. If has an eigenvalue at the origin, then all other eigenvalues must have magnitude at least .The bound in Corollary 2.3 implies, under the assumptions, that the matrix is near normal, in the sense of having a well-conditioned eigenvector matrix. To see this, suppose that

(8) |

Write in the following form:

where is the th column of and is the th row of . Now choose a polynomial such that and for . Applying inequality (8) to , we see that , whence

If each column of is taken to be of -norm , then each row of has -norm at most . Therefore, the Frobenius norm of is at most and the Frobenius norm of is at most . Since the operator norm of a matrix is less than or equal to the Frobenius norm, we have

(9) |

where is the quantity defined in (5). Thus, under the assumptions of Corollary 2.3, . Actually, a somewhat stronger relation is known between the best-conditioned eigenvector matrix in the operator norm and the best-conditioned eigenvector matrix in the Frobenius norm. It is shown in [36] that

where is the operator norm condition number and is the Frobenius norm condition number. It follows that inequality (9) can be replaced by

and if , then .

In fact a stronger bound on may be given when we interpolate with Blaschke products instead of polynomials. This estimate relates directly to the separation constant . It requires a more general version of von Neumann’s inequality for holomorphic functions which follows from the same approximation argument already used above. The finite Blaschke products of degree defined by

are the minimal norm interpolants that are at and at the other eigenvalues of . At the spectrum of they attain the same values as the , and hence . Since is a contraction, the generalized von Neumann’s inequality yields

Arguing the same way as above, we get that the Frobenius norm of is at most and that of is at most ; thus the condition number of (in either the Frobenius norm or the operator norm) satisfies

(10) |

Using the result in [36], we can subtract from the right-hand side of (10) to obtain a stronger bound on .

### 2.2 Bounds via Condition Numbers

As before, let be an matrix with distinct eigenvalues and decomposition . If the quantity from (5) satisfies , then Crouzeix’s conjecture immediately holds for . In general, can be arbitrarily large. However, it is possible to obtain bounds on in the important case where is a matrix representation of a compressed shift . For additional background material concerning compressed shifts and their matrix representations, we refer the reader to [19] and Chapter in [17].

To that end, let be a finite Blaschke product with distinct zeros and let denote a single Blaschke factor. A useful basis of is the Takenaka-Malmquist basis^{2}^{2}2The name of this basis is not standard. According to [19] these appeared in Takenaka’s 1925 paper, [38]. The text [32] discusses this basis for the case including infinite Blaschke products and uses the term “Malmquist-Walsh” basis., defined as follows

Writing with respect to the Takenaka-Malmquist basis gives the matrix representation where

For example, if , then

Let be the diagonal matrix with diagonal entries . If has distinct zeros, then has distinct eigenvalues and so can be written as for some matrix . Here are tractable formulas for and . The proof appears in the appendix in Section 5.

###### Theorem 2.4.

Let be a finite Blaschke product with distinct zeros . Let be the matrix representation of with respect to the Takenaka-Malmquist basis. Then , where is the diagonal matrix with for and the entries of and are given by

###### Remark 2.5.

This theorem gives an initial bound on the condition number of , namely

If the zeros of are sufficiently separated (in the Euclidean and pseudohyperbolic metrics) and at least are sufficiently near the unit circle , then the formulas in Theorem 2.4 show that we can make the off-diagonal entries of and arbitrarily close to and hence the condition number of arbitrarily close to .

Using the formulas for and , we can also obtain the following bound on the condition number of . Note that this also provides a bound on for the matrix representation of the compressed shift .

###### Theorem 2.6.

As the bound in Theorem 2.6 does not depend on , it seems better than the bound in (10) in situations where is large. Nevertheless, it includes significant dependence on , and the appearance of the constant prevents this estimate from being sharp as . These concerns motivate Question 1.1, which was posed in the introduction.

To prove Theorem 2.6, we need the following lemma, which is likely well known.

###### Lemma 2.7.

Let be a finite Blaschke product with distinct zeros , and let denote the separation constant of the zeros of given in (7). Define for and let be the Gramian matrix defined by . Then

###### Proof.

###### Proof of Theorem 2.6.

We use the estimate where denotes the numerical radius. Then fixing with , we have

(11) |

For any , define the functions

Then (11) can be rewritten as:

where we used the fact that if , then and if , then . Furthermore, observe that each This and Lemma 2.7 give:

This shows that and an analogous argument gives the same bound for . ∎

## 3 Examples of Extremal Functions and Vectors

Let be a bounded simply connected domain with smooth boundary containing the spectrum of an matrix . We consider functions , the closed unit ball in , for which , taken over all , is attained. Recall that such a function is called *extremal* for and if, furthermore, is a non-zero vector where , then is called an associated *extremal* vector. As discussed in the introduction, the study of such functions is closely related to recent proofs and investigations of Crouzeix’s conjecture.

One can also measure the size of via its numerical radius. Given , we say that is *-extremal*, if is a function for which , taken over all , is attained. A vector is an associated *-extremal* vector if

In this section, we consider two classes of examples of extremal functions and vectors.

### 3.1 Compressions of the Shift with

Let be a finite Blaschke product. Recall that and the compression of the shift with symbol is defined by , where is the orthogonal projection of onto and is the shift operator. In [18, Theorem 2, pp. 22], Garcia and Ross showed that the extremal functions for are exactly the finite Blaschke products with . We encode their result in the following theorem:

###### Theorem 3.1.

Let be a finite Blaschke product with and let . Then . Moreover if and only if is a finite Blaschke product with

Here we present a simple proof of this result when has distinct zeros .

###### Proof.

First, fix . Then von Neumann’s inequality implies that

Now fix with . Then by [34, Proposition 5.1], there is a unique such that and , where denotes multiplication by . Moreover, is a finite Blaschke product of degree at most . Sarason’s work [34, pp. 187] implies that for . Since the , the interpolation problem has a unique solution in (see [1, pp. 77, Lemma 6.19]) and so,

Similarly, if we begin with a Blaschke product of