Spectra of upper triangular operator matrices with m-complex symmetric operator entries along the main diagonal

摘要

Abstract Let H be a separable complex Hilbert space, a bounded linear operator T:H⟶Hdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$T:Hlongrightarrow H$$end{document} is m-complex symmetric if there exists a conjugation J on H such that Δm(T)=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta _{m}(T)=0$$end{document} where Δm(T)=∑k=0m(-1)m-kmkT∗kJTm-kJ.documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta _{m}(T)=sum _{k=0}^{m} (-1)^{m-k}left( {begin{array}{c}m kend{array}}right) T^{*k}JT^{m-k}J.$$end{document} In this paper, we study some spectral properties of m-complex symmetric operators like finite ascent and polaroid conditions. As a consequence, we examine Weyl type theorems when the m-complex symmetric operator T is polaroid. We also study the transition of spectral properties between two upper triangular operator matrices for which the diagonal entries are m-complex symmetric.