Searcharxiv⌕ Search

arXiv subjects

Ilya M. Spitkovsky

Publications and source records attributed to Ilya M. Spitkovsky.

At least 19 recordsLinked to original sources
arXiv2026

The wanted extension of Fujii and Tsurumaru's formula for the spectral radius of the Bell-CHSH operator

This paper is motivated by a recent paper of Yuki Fujii and Toyohiro Tsurumaru in which they established a beautiful formula for the spectral radius of the Bell-CHSH operator on finite-dimensional Hilbert spaces. To tackle the operator on infinite-dimensional spaces, they elaborated a method based on appropriate approximation of commutators of infinite-dimensional orthogonal projections by commutators of orthogonal projections on finite-dimensional spaces. We here give a proof of Fujii and Tsurumaru's original formula that works in all dimensions. We also present an alternative approximation procedure, uncover the connection of the problem with block Toeplitz operators, and derive good estimates and explicit expressions for the spectral radius in concrete cases.

math.FA↗
arXiv2026

On Kippenhahn curves of low rank partial isometries

Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration.

math.FA↗
arXiv2024

Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

Reciprocal matrices are tridiagonal matrices $(a_{ij})_{i,j=1}^n$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For these matrices, criteria are established under which their Kippenhahn curves contain elliptical components or even consist completely of such. These criteria are in terms of system of homogeneous polynomial equations in variables $(\left|a_{j,j+1}\right|-\left|a_{j+1,j}\right|)^2$, and established via a unified approach across arbitrary dimensions. The results are illustrated, and specific numerical examples provided, for $n=7$ thus generalizing earlier work in the lower dimensional setting.

math.FA↗
arXiv2022

On low-dimensional partial isometries

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 $n=5$.

math.FA↗
arXiv2022

Numerical ranges of Foguel operators revisited

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 their LAA'2021 paper that $W(F_{aI})$ for non-zero $a\in\mathbb C$ might be an elliptical disk. In this paper, we described $W(F_{aI})$ explicitly and, as it happens, it is not.

math.FA↗
arXiv2021

On partial isometries with circular numerical range

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 shift operator $S_B$ generated by a finite Blaschke product $B$. We then use the description of the numerical range of $S_B$ as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

math.FA↗
arXiv2021

On multivariable matrix spectral factorization method

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 important due to its practical applications. Therefore, several authors have developed algorithms for factorization. The Janashia-Lagvilava algorithm is a relatively new method for matrix spectral factorization which has proved to be useful in several applications. In this paper, we extend this method to the multivariable case. Consequently, a new numerical algorithm for multivariable matrix spectral factorization is constructed.

math.CV↗
arXiv2021

Low-dimensional reciprocal matrices with elliptical components of their Kippenhahn curves

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 corollary, we also provide a complete description of higher-rank numerical ranges when the criteria are met.

math.FA↗
arXiv2021

The Gau-Wu Number for $4\times 4$ and Select Arrowhead Matrices

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 $x_j$ such that the scalar products $\langle Ax_j,x_j\rangle$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4\times4$ matrices according to the values of their Gau--Wu numbers.

math.FA↗
arXiv2021

The norm attainment problem for functions of projections

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 norm.

math.FA↗
arXiv2020

Kippenhahn curves of some tridiagonal matrices

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.

math.FA↗
arXiv2020

On some 4-by-4 matrices with bi-elliptical numerical ranges

A complete description of 4-by-4 matrices $\begin{bmatrix}αI & C \\D & βI\end{bmatrix}$, 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. In particular cases when in addition $α-β$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.

math.FA↗
arXiv2020

On the Stampfli point of some operators and matrices

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 $λ\in\mathbb C$ delivering the minimum value of the norm of $A-λI$. We derive some results concerning the location of St($A$) for several classes of operators, including 2-by-2 block operator matrices with scalar diagonal blocks and 3-by-3 matrices with repeated eigenvalues. We also show that for almost normal $A$ its Stampfli point lies in the convex hull of the spectrum, which is not the case in general. Some relations between the property St($A$)=0 and Roberts orthogonality of $A$ to the identity operator are established.

math.FA↗
arXiv2020

On the numerical range of some block matrices with scalar diagonal blocks

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.

math.FA↗
arXiv2019

Singularities of Base Polynomials and Gau-Wu Numbers

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 $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu numbers for $3\times 3$ matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified $W(A)$ for $4\times 4$ unitarily irreducible $A$ with irreducible base curve according to singularities of that curve. When $A$ is a unitarily irreducible $n\times n$ matrix, we give necessary conditions for $k(A) = 2$, characterize $k(A) = n$, and apply these results to the case of unitarily irreducible $4\times 4$ matrices. However, we show that knowledge of the singularities is not sufficient to determine $k(A)$ by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different $k(A)$. In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible $A$ with reducible base curve and show that we can find corresponding matrices with identical base curve but different $k(A)$. Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky, generalizing their result in the process.

math.FA↗