Ilya Spitkovsky

New York University Abu Dhabi · Science and Mathematics

We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. It can be used to construct $n\times n$ orthogonal matrices in $Z_p$ with very high values of $n$ and $p$, and also orthogonal matrices with a certain circulant structure. Equally well one can construct paraunitary filter banks or wav...

Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $n\leq 4$ but fail starting with $n=5$.

A Hilbert space operator A is said to be core invertible if it has an inner inverse whose range coincides with the range of A and whose null space coincides with the null space of the adjoint of A . This notion was introduced by Baksalary, Trenkler, Rakić, Dinčić, and Djordjević in the last decade, who also proved that core invertibility is equival...

An efficient method for construction of J-unitary matrix polynomials is proposed, associated with companion matrix functions the last row of which is a polynomial in 1/t. The method relies on Wiener-Hopf factorization theory and stems from recently developed J-spectral factorization algorithm for certain Hermitian matrix functions.

Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $n\leq 4$ but fail starting with $n=5$.

The question of which elements of the open unit disc occur as eigenvalues of n-by-n doubly stochastic matrices has long been open, in spite of a number of intriguing partial results. By enhancing a natural, but slightly false, conjecture, we gain some new computational insights into the problem. We also apply the classical field of values to give s...

Spectral factorization is a prominent tool with several important applications in various areas of applied science. Wiener and Masani proved the existence of matrix spectral factorization. Their theorem has been extended to the multivariable case by Helson and Lowdenslager. Solving the problem numerically is challenging in both situations, and also...

The Foguel operator is defined as $F_T=\begin{bmatrix}S^* & T \\ 0 & S\end{bmatrix}$, where $S$ is the right shift on a Hilbert space $\mathcal H$ and $T$ can be an arbitrary bounded linear operator acting on $\mathcal H$. Obviously, the numerical range $W(F_0)$ of $F_T$ with $T=0$ is the open unit disk, and it was suggested by Gau, Wang and Wu in...

A key ingredient of Janashia–Lagvilava matrix spectral factorization method, which is a theorem from complex function theory, is generalized to pure algebraic form.

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds for $n=5$.

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj...

Janashia-Lagvilava algorithm is a relatively new method of matrix spectral factorization. In our previous publications on this topic, we demonstrated that the algorithm is capable to compete with other existing methods of factorization. In the present paper, we provide further refinements of the algorithm emphasizing that it might have a significan...

In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ ⁿ cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n . The proof is based on the unitary similarity of A to a compressed shift operator S B generated by...

By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+1,i= 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For...

In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry $A$ acting on $\mathbb{C}^n$ cannot have a circular numerical range with a non-zero center, and proved this conjecture for $n\leq 4$. We prove it for operators with $\mathrm{rank}\,A=n-1$ and any $n$. The proof is based on the unitary similarity of $A$ to a compressed shi...

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.

The 4-by-4 nilpotent matrices whose numerical ranges have nonparallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.

The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. Every skew projection on that Hilbert space is contained in such an algebra and hence the results of the paper also describe functions of skew projections and their adjoints that...

Spectral factorization is a prominent tool with several important applications in various areas of applied science. Wiener and Masani proved the existence of matrix spectral factorization. Their theorem has been extended to the multivariable case by Helson and Lowdenslager. Solving the problem numerically is challenging in both situations, and also...

By definition, reciprocal matrices are tridiagonal $n$-by-$n$ matrices $A$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For $n\leq 6$, we establish criteria under which the numerical range generating curves (also called Kippenhahn curves) of such matrices consist of elliptical components only. As a corollar...

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau--Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors...

The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.

The problems of matrix spectral factorization and J-spectral factorization appear to be important for practical use in many MIMO control systems. We propose a numerical algorithm for J-spectral factorization which extends Janashia-Lagvilava matrix spectral factorization method to the indefinite case. The algorithm can be applied to matrices that ha...

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the...

Tridiagonal matrices with constant main diagonal and reciprocal pairs of off-diagonal entries are considered. Conditions for such matrices with sizes up to 6-by-6 to have elliptical numerical ranges are obtained.

A complete description of 4-by-4 matrices α I C D β I , with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which C − D ∗ also is a scalar multiple of the identity. In particular cases when in addition α − β...

The 4-by-4 nilpotent matrices the numerical ranges of which have non-parallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.

Tridiagonal matrices with constant main diagonal and reciprocal pairs of off-diagonal entries are considered. Conditions for such matrices with sizes up to 6-by-6 to have elliptical numerical ranges are obtained.

This perspective originated during the Isaac Newton Institute for Mathematical Sciences research programme “Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications (WHT)”. It fuelled intensive discussions on current pure and applied mathematical challenges of the Wiener-Hopf technique, driven...

This perspective originated during the Isaac Newton Institute for Mathematical Sciences research programme “Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications (WHT)”. It fuelled intensive discussions on current pure and applied mathematical challenges of the Wiener-Hopf technique, driven...

A complete description of 4-by-4 matrices $\begin{bmatrix}\alpha I & C \\D & \beta I\end{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity....

The center of mass of an operator $A$ (denoted St($A$), and called in this paper as the {\em Stampfli point} of A) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique $\lambda\in\mathbb C$ delivering the minimum value of the norm of $A-\lambda I$. We derive some results concerning the location of St($A$) for several classes...

In this paper, we provide a biography of Professor Rajendra Bhatia and discuss some of his influential mathematical works as one of the leading researchers in matrix analysis and linear algebra.

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive inv...

Several new verifiable conditions are established for matrices of the form αIn−kCDβIk to have the numerical range equal the convex hull of at most k ellipses. For k = 2, these conditions are also necessary, provided that the ellipses are co-centred.

In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n×n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). We refer to k(A) as the Gau–Wu number of the matrix A. In this paper we take an algebraic geometric approach and co...

Several new verifiable conditions are established for block matrices with scalar diagonal blocks to have the numerical range equal the convex hull of at most k ellipses where k by k is the size of the smaller diagonal block. For k = 2, these conditions are also necessary, provided that the ellipses are co-centered.

The notion of the \(\mathcal L\)-convolution operator is introduced by changing the Fourier operator in the definition of the (regular) convolution operator to the operator intertwining the Sturm-Liouville operator \(\mathcal L\) with the multiplication operator. Along the same lines, the \(\mathcal L\)-Wiener-Hopf operator is introduced. For the l...

Wiener-Granger causality (WGC) measures causal influence between two stationary stochastic time series X n and Y n , n = 1, 2,. . ., based on the statistical predictability of one depending on another. Namely, if X n = ∞ k=1 a k X n−k + ε 1 (n) and X n = ∞ k=1 b k X n−k + ∞ k=1 c k Y n−k + ε 2 (n) are autoregressive representations of X solely and...

In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $n\times n$ matrix $A$, which counts the maximal number of orthonormal vectors $\textbf x_j$ such that the scalar products $\langle A\textbf x_j,\textbf x_j\rangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A...

The criterion is obtained for operators A from the algebra generated by two orthogonal projections P, Q to have a compatible range, i.e., coincide with A* on the orthogonal complement to the sum of the kernels of A and A*. In the particular case of A being a polynomial in P, Q, some easily verifiable conditions are derived.

We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same ord...

We describe continuity properties of the multivalued inverse of the numerical range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove in particular that $f_A^{-1}$ is strongly continuous at all points of the interior of the numerical range $W(A)$. We...

The notions of the L-convolution operator and the ℒ-Wiener–Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm–Liouville operator on the axis ℒ. In the case of the zero potential, the introduced operators coincide with the convolution op...

Let u be a Hermitian involution, and e an orthogonal projection, acting on the same Hilbert space \(\mathcal{H}\). We establish the exact formula, in terms of \(||{eue}||\), for the distance from e to the set of all orthogonal projections q from the algebra generated by e, u, and such that quq = 0.

The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in partic...

Let G be a block matrix function with one diagonal block A being positive definite and the off diagonal blocks complex conjugates of each other. Conditions are obtained for G to be factorable (in particular, with zero partial indices) in terms of the Schur complement of A.

The maximal numerical range W 0 (A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A * A corresponding to its maximal eigenvalue. So, always W 0 (A) ⊆ W (A). Conditions under which W 0 (A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W 0 (A) = W (A)....

The meta theorem of this paper is that Halmos’ two projections theorem is something like Robert Sheckley’s Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supers...

The spectral factorization mapping $F\to F^+$ puts a positive definite integrable matrix function $F$ having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function $F^+$ such that $F = F^+(F^+)^*$ almost everywhere. The main question addressed here is to what extent $\|F^+ - G^+\|_{H_2}$ is controlled by...

We present a certain generalization of Smirnov’s theorem on functions from the Hardy spaces Hp{H_{p}} . We provide some applications of the proposed generalization. Namely, we give an equivalent characterization of outer analytic rectangular matrix functions, and give a simple proof of the uniqueness of spectral factorization of rank deficient matr...

We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given.

The higher rank numerical range is described for a class of matrices which happen to be unitarily reducible to direct sums of (at most) 2-by-2 blocks. In particular, conditions are established under which tridiagonal matrices have elliptical rank-k numerical ranges.

The present article covers the contributions of the speakers at the memorial session “Remembering Leiba Rodman” at IWOTA 2015. © Springer International Publishing AG, part of Springer Nature 2018.

For a triangular n-by-n matrix function with a fixed ordered set of the indices of its diagonal entries, an explicit description is obtained for all possible n-tuples of the partial indices. © Springer International Publishing AG, part of Springer Nature 2018.

For a given $n$-by-$n$ matrix $A$, its {\em normalized numerical range} $F_N(A)$ is defined as the range of the function $f_{N,A}\colon x\mapsto (x^*Ax)/(\norm{Ax}\cdot\norm{x})$ on the complement of $\ker A$. We provide an explicit description of this set for the case when $A$ is normal or $n=2$. This extension of earlier results for particular ca...

Given two orthogonal projections P and Q, we are interested in all unitary operators U such that UP=QU and UQ=PU. Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the r...

This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the...

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case...

One Spring day of 1984, Mark Grigor’evich Krein showed me the abstract of a PhD thesis he received in the mail from Rostov-on-Don University. He liked it at first glance, so we set to read it together more carefully, and towards the end of this reading he decided to write a formal (enthusiastically positive) report to the PhD defence committee.

In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are paral...

Let u be a hermitian involution, and e an orthogonal projection, acting on the same Hilbert space. We establish the exact formula, in terms of the norm of eue, for the distance from e to the set of all orthogonal projections q from the algebra generated by e,u, and such that quq=0.

The normalized numerical range of an operator A is defined as the set FN(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that FN(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shift...

Predicting quantum phase transitions by signatures in finite models has a long tradition. Here we consider the numerical range $W$ of a finite dimensional one-parameter Hamiltonian, which is a planar projection of the convex set of density matrices. We propose the new geometrical signature of non-analytic points of class $C^2$ on the boundary of $W...

An elementary proof of Robinson’s Energy Delay Theorem on minimum-phase functions is provided. The situation in which the energy conservation property holds for an infinite number of lags is fully described.

We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence co...

For a given Laurent polynomial matrix function $S$, which is positive definite on the unit circle in the complex plane, we consider all possible polynomial spectral factors of $S$ which are not necessarily invertible inside the unit circle.

We consider three different ways of algorithmization of the Janashia-Lagvilava spectral factorization method. The first algorithm is faster than the second one, however, it is only suitable for matrices of low dimension. The second algorithm, on the other hand, can be applied to matrices of substantially larger dimension. The third algorithm is a s...

Matrices subordinate to trees are considered. An efficient normality characterization for any such matrix is given, and several consequences (not valid for general normal matrices) of it are established. In addition, the existence (and enumeration) of flat portions on the boundary of the field of values of matrices subordinate to a tree is characte...

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $...

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discont...

In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are paral...

Tridiagonal matrices are considered for which the main diagonal consists of zeroes, the sup-diagonal of all ones, and the entries on the sub-diagonal form a geometric progression. The criterion for the numerical range of such matrices to have line segments on its boundary is established, and the number and orientation of these segments is described...

We solve a Riemann–Hilbert problem with almost periodic coefficient G, associated to a Toeplitz operator in a class which is closely connected to finite interval convolution equations, based on a generalization of the so-called table method. The explicit determination of solutions to that problem allows one to establish necessary and sufficient con...

This note corrects a theorem characterizing the points where weak continuity can fail for the inverse numerical range map.

For certain tridiagonal matrices of small size, we give a complete description of flat portions on the boundary of their numerical range. We also discuss the conditions for these numerical ranges to be elliptical.

A simple constructive proof of polynomial matrix spectral factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.

We consider defined on the real line ℝ matrix functions with monomial terms of the form ceiλx on the main diagonal and one row, and with zero entries elsewhere. The factorability of such matrices is established and, moreover, the algorithm for their factorization is provided. In particular, formulas for the partial indices are derived, and conditio...

A necessary condition for the existence of spectral factorization is positive definiteness a.e. on the unit circle of a matrix function which is being factorized. Correspondingly, the existing methods of approximate computation of the spectral factor can be applied only in the case where the matrix function is positive definite. However, in many pr...

Factorizations of the Wiener–Hopf type of classes of matrix functions with various symmetries are studied, in the abstract context of Banach algebras of functions over connected abelian compact groups. The symmetries in question are induced by involutive automorphisms or antiautomorphisms of the general linear group, and include many symmetries stu...

This paper considers matrices A is an element of M-n(C) whose numerical range contains boundary points generated by multiple linearly independent vectors. Sharp bounds for the maximum number of such boundary points (excluding flat portions) are given for unitarily irreducible matrices of dimension <= 5. An example is provided to show that there may...

Canonical factorization criterion is established for a class of block triangular almost periodic matrix functions. Explicit factorization formulas are also obtained, and the geometric mean of matrix functions in question is computed.

Conditions are established under which Fredholmness, Coburn's property and one- or two-sided invertibility are shared by a Toeplitz operator with matrix symbol $G$ and the Toeplitz operator with scalar symbol $\det G$. These results are based on one-sided invertibility criteria for rectangular matrices over appropriate commutative rings and related...

For a given n×nn×n matrix A, let k(A)k(A) stand for the maximal number of orthonormal vectors xjxj such that the scalar products 〈Axj,xj〉〈Axj,xj〉 lie on the boundary of the numerical range W(A)W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau–Wu number of the matrix A. We compute k(A)k(A) for two classes of n×...

It is shown that every matrix in a large class of n-by-n doubly cyclic Z+Z+ matrices with negative determinant has exactly one eigenvalue in the closed left half-plane. This generalizes a result for n=4n=4 used in a recent analysis of cancer cell dynamics. A further conjecture is made based on computational evidence. All work relates to the inertia...

Possible shapes of numerical ranges of rank-two operators are studied. In particular it is proved that for 4-by-4 unitarily irreducible matrices with an eigenvalue of geometric multiplicity two, the numerical ranges have at most one flat portion on the boundary and there are no multiply generated round boundary points.

A classification of all possible shapes is given for numerical ranges of doubly stochastic matrices. The tests determining the shape are also provided, along with illustrating examples.

We show that the kernel and/or cokernel of a block Toeplitz operator T (G) are trivial if its matrix-valued symbol G satisfies the condition \(G(t^{-1})G(t)^*\;=\;I_N\). As a consequence, the Wiener–Hopf factorization of G (provided it exists) must be canonical. Our setting is that of weighted Hardy spaces on the unit circle. We extend our result t...

Let PP and QQ be idempotents in a real or complex associative algebra and consider the list of products P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,…P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,…. The number of factors is called the order of the product. We say that PP and QQ are tightly coupled if the list contains two products which take the same value and whose orders differ by...