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In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the

In this paper, the authors study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. They first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. They propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. They also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. The authors then extend the H^\\infty-functional calculus to an \\overline{H^\\infty}+H^\\infty-functional calculus for the compressions of the shift. Next, the authors consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos's Problem 5 and then establish a matrix-valued version of Abrahamse's Theorem. They also solve a subnormal Toeplitz completion problem of 2\\times 2 partial block Toeplitz matrices. Further, they establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs.

This work explores joint hyponormality of Toeplitz pairs. Topics include: hyponormality of Toeplitz pairs with one co-ordinate a Toeplitz operator with analytic polynomial symbol; hyponormality of trigonometric Toeplitz pairs; and the gap between $2$-hyponormality and subnormality.

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ( SP ) (related to the Aluthge transform) and the Square Root Problem ( SRP ) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ( SP ) and ( SRP ) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For p -atomic measures with p ≤ 6 , our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a 7-atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by Exner (J Oper Theory 61:419–438, 2009), and to a recent conjecture formulated by Curto et al. (Math Nachr 292:2352–2368, 2019).

Let T∈B(H)T\\in \\mathcal {B}(\\mathcal {H}) be a bounded linear operator on a Hilbert space H\\mathcal {H}, and let T≡V|T|T \\equiv V|T| be the polar decomposition of TT. The mean transform of TT is defined by T^:=12(V|T|+|T|V)\\widehat {T}:=\\frac {1}{2}(V|T|+|T|V). In this paper we study the iterates of the mean transform and we define the mean limit of an operator as the limit (in the operator norm) of those iterates. We obtain new estimates for the numerical range and numerical radius of the mean transform in terms of the original operator. For the special class of unilateral weighted shifts we describe the precise relationship between the spectral radius and the mean limit, and obtain some sharp estimates.

In this paper the notion of slantification of a Hankel operator  on the space H 2 ( T n )  , the Hardy space of n -torus, is introduced. Various properties including hyponormality, isometric behaviour, co-isometric behaviour and compactness of these operators are also studied.

An outline of Jörg Eschmeier’s main mathematical contributions is organized both on a historical perspective, as well as on a few distinct topics. The reader can grasp from our essay the dynamics of spectral theory of commutative tuples of linear operators during the last half century. Some clear directions of future research are also underlined.

In this book, the authors introduce a matricial approach to the truncated complex moment problem and apply it to the case of moment matrices of flat data type, for which the columns corresponding to the homogeneous monomials in $z$ and $\\bar z$ of highest degree can be written in terms of monomials of lower degree. Necessary and sufficient conditions for the existence and uniqueness of representing measures are obtained in terms of positivity and extension criteria for moment matrices.

In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\\mu$ in the Truncated Complex Moment Problem: $\\gamma_{ij}=\\int \\bar z^iz^j\\, d\\mu$ $(0\\1e i+j\\1e 2n)$. Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix $M(n)(\\gamma)$ associated with $\\gamma \\equiv \\gamma ^{(2n)}$: $\\gamma_(00), \\dots, \\gamma_{0,2n},\\dots, \\gamma _{2n,0}$, $\\gamma_{00}>0$.This study includes new conditions for flat (i.e., rank-preserving) extensions $M(n+1)$ of $M(n)\\ge 0$; each such extension corresponds to a distinct rank $M(n)$-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations.In a variety of applications, including cases of the quartic moment problem ($n=2$), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.

We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.