Ilya M. Spitkovsky

The numerical range of 3 × 3 matrices

Abstract Let A be an n × n complex matrix. Then the numerical range of A, W(A), is defined to be { x∗Ax: x ∈ C n , x∗x = 1} . In this article a series of tests is given, allowing one to determine the shape of W(A) for 3 × 3 matrices. Reconstruction of A, up to unitary similarity, from W(A) is also examined.

Singular integral operators with PC symbols on the spaces with general weights

We prove a Fredholm criterion for singular integral operators with piecewise continuous coefficients on Lp spaces with the weight ϱ satisfying the Hunt-Muckenhoupt-Wheeden condition. The proof is based on the case of power weight ϱ(t) = Π ¦t − tj¦βj that was investigated by Gohberg and Krupnik more than 20 years ago.

An Overview of Matrix Factorization Theory and Operator Applications

These lecture notes present an extensive review of the factorization theory of matrix functions relative to a curve with emphasis on the developments of the last 20–25 years. The classes of functions considered range from rational and continuous matrix functions to matrix functions with almost periodic or even semi almost periodic entries. Also included are recent results about explicit factorization based on the state space method from systems theory, with examples from linear transport theory. Related applications to Riemann-Hilbert boundary value problems and the Fredholm theory of various classes of singular integral operators are described too. The applications also concern inversion of singular integral operators of different types, including Wiener-Hopf and Toeplitz operators.

Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present nece ssary and sufficient conditions for the existence of extensions in terms of Toeplit;z and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.

Banach Algebras Generated by N Idempotents and Applications

It is well known that for Banach algebras generated by two idempotents and the identity all irreducible representations are of order not greater than two. These representations have been described completely and have found important applications to symbol theory. It is also well known that without additional restrictions on the idempotents these results do not admit a natural generalization to algebras generated by more than two idempotents and the identity. In this paper we describe all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations. These representations are of order not greater than N and allow us to construct a symbol theory with applications to singular integral operators.

Once more on algebras generated by two projections

We consider von Neumann algebras generated by two arbitrary orthoprojections on a Hilbert space. A canonical decomposition is obtained for elements A of these algebras in terms of the operator angle between the ranges of the above-mentioned projections. This decomposition leads to explicit descriptions and formulas for kernels, ranges, spectra and essential spectra, (generalized) inverses, and other objects related to A.

(Semi)-Fredholmness of Convolution Operators on the Spaces of Bessel Potentials

The consideration of above mentioned operators on the union of intervals and/or rays is reduced to the canonical situation of operators W k on L P (ℝ+) with semi almost periodic presymbols K at the expense of inflating the size of K. The Fredholm theory (that is, conditions of n-, d-normality and the index formula) is developed. In particular, relations between (semi-)Fredholmness of W K , invertibility of \({W_{{k_ \pm }}}\) with K ± being almost periodic representatives of K at ±∞, and factorability of K ± are established.

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceLp(Γ,ρ) with an arbitrary weight ρ is studied. The contour Γ consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.

On a new algorithm for almost periodic factorization

An algorithm is proposed allowing to find necessary and sufficient conditions for existence of an almost periodic factorization for several new classes of triangular and block triangular matrix functions.

Pre-images of boundary points of the numerical range

This paper considers matrices A ∈ Mn(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 be infinitely many for n = 6 . For matrices unitarily similar to tridiagonal, however, a finite upper bound is found for all n . A somewhat unexpected byproduct of this is an explicit example of A ∈ M5(C) which is not tridiagonalizable via a unitary similarity. Mathematics subject classification (2010): Primary 15A60, 47A12; Secondary 54C08.