Raúl E Curto

Given a bounded linear operator T with canonical polar decomposition T≡VT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$T \equiv V\left |T\right |$$ \end{document}, the Aluthge transform of T is the operator Δ(T):=TVT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta (T):=\sqrt {\left |T\right |} V \sqrt {\left |T\right |}$$ \end{document}. For P an arbitrary positive operator such that V P = T, we define the extended Aluthge transform of T associated with P by ΔP(T):=PVP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta _P(T):=\sqrt {P} V \sqrt {P}$$ \end{document}. First, we establish some basic properties of ΔP; second, we study the fixed points of the extended Aluthge transform; third, we consider the case when T is an idempotent; next, we discuss whether ΔP leaves invariant the class of complex symmetric operators. We also study how ΔP transforms the numerical radius and numerical range. As a key application, we prove that the spherical Aluthge transform of a commuting pair of operators corresponds to the extended Aluthge transform of a 2 × 2 operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.

We first discuss the spherical Aluthge and spherical Duggal transforms for commuting pairs of operators on Hilbert space. Second, we study the fixed points of these transforms, which are the spherically quasinormal commuting pairs. In the case of commuting 2-variable weighted shifts, we prove that spherically quasinormal pairs are intimately related to spherically isometric pairs. We show that each spherically quasinormal 2-variable weighted shift is completely determined by a subnormal unilateral weighted shift (either the 0-th row or the 0-th column in the weight diagram). We then focus our attention on the case when this unilateral weighted shift is recursively generated (which corresponds to a finitely atomic Berger measure). We show that in this case the 2-variable weighted shift is also recursively generated, with a finitely atomic Berger measure that can be computed from its 0-th row or 0-th column. We do this by invoking the relevant Riesz functionals and the functional calculus for the columns of the associated moment matrix.

Over the last fifty years, several generations of operator theorists have been acquainted with the expository writings of Paul R. Halmos, John B. Conway, Ronald G. Douglas and Allen P. Shields. A common theme in these works is the ubiquity of unilateral weighted shifts acting on the Hilbert space ℓ 2(ℤ) and ℓ 2(ℤ+), or its function-theoretic counterparts, the multiplication operators on the Hardy space, Bergman space, Dirichlet space, weighted Bergman spaces and more generally reproducing kernel Hilbert spaces. While weighted shifts represent the core of the celebrated survey article by Allen P. Shields [80], they are also explicitly mentioned in at least 29 of the 250 problems listed by P.R. Halmos in [65] (see also [64]. Weighted shifts have been, and continue to be, the source of countless examples and counterexamples in spectral theory, invariant subspace theory, C * -dynamical system theory, and subnormal and hyponormal operator theory and its generalizations. Here’s a sample elementary result.

We give a brief survey of subnormality and hyponormality of Toeplitz operators on the vector-valued Hardy space of the unit circle. We also solve the following subnormal Toeplitz completion problem: Complete the unspecified rational Toeplitz operators (i.e., the unknown entries are rational Toeplitz operators) of the partial block Toeplitz matrixG:=Tω1¯??Tω2¯(ω1andω2are finite Blaschke products)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ G: = \left[ \begin{array}{ccc} {{T_{\overline {{\omega _1}} }}}&? \\ ? &{{T_{\overline {{\omega _2}} }}} \end{array} \right]({\omega _1}\,and\,{\omega _2}\,\text{are finite Blaschke products)} $$ \end{document} to make G subnormal.