ACTA issues

Weyl functions of bounded quasi-selfadjoint operators and block operator Jacobi matrices

Yury Arlinskiń≠, Lutz Klotz

Acta Sci. Math. (Szeged) 76:3-4(2010), 585-626

Abstract. A bounded operator $T$ in a separable Hilbert space ${\eufm H}$ is called quasi-selfadjoint if $\ker(T-T^*)\not=\{0\} $ and ${\eufm N}$-quasi-selfadjoint if ${\eufm N}\supseteq{\rm ran }(T-T^*)$, where ${\eufm N}$ is a subspace of ${\eufm H}$. An ${\eufm N}$-quasi-selfadjoint operator $T$ is called ${\eufm N}$-simple if the linear hull of $\{T^n{\eufm N}, n=0,1,\ldots\} $ is dense in ${\eufm H}$. We study the ${\eufm N}$-Weyl function $M(z)=P_{\eufm N}(T-zI_{\eufm H})^{-1}{\mathrel{|^{\kern -2pt\scriptscriptstyle\setminus }} }{\eufm N}$ of an ${\eufm N}$-quasi-selfadjoint operator and define its so-called ``Schur parameters". The main result of the paper is that any ${\eufm N}$-quasi-selfadjoint and ${\eufm N}$-simple operator is unitarily equivalent to an operator given by a special block operator Jacobi matrix constructed by means of the Schur parameters of its ${\eufm N}$-Weyl function.

AMS Subject Classification (1991): 47A45, 47A48, 47A56, 47B36

Keyword(s): quasi-selfadjoint operator, ${\eufm N}$-simple operator, the Weyl function, the Schur transformation, the Schur parameters, block Jacobi operator matrix

Received February 20, 2009. (Registered under 36/2009.)