Complex isosymmetric operators

Document Type: Original Article


1 Kanagawa University

2 Department of Mathematics and Statistics‎, ‎Sejong University‎, ‎Seoul 143-747‎, ‎Korea

3 Department of Mathematics‎, ‎Cochin university of Science and Technology‎, ‎Kochi‎, ‎India

4 Department of Mathematics‎, ‎Tohoku Medical and Pharmaceutical University‎, ‎Sendai 981-8558‎, ‎Japan


‎In this paper‎, ‎we introduce complex isosymmetric and $(m,n,C)$-isosymmetric operators on a Hilbert space $\mathcal H$ and study properties of such operators‎. ‎In particular‎, ‎we prove that if $T \in {\mathcal B}(\mathcal H)$ is an $(m,n,C)$-isosymmetric operator and $N$ is a $k$-nilpotent operator such that $T$ and $N$ are $C$-doubly commuting‎, ‎then $T‎ + ‎N$ is an $(m+2k-2‎, ‎n+2k-1,C)$-isosymmetric operator‎. ‎Moreover‎, ‎we show that if $T$ is $(m,n,C)$-isosymmetric and if $S$ is $(m',D)$-isometric and $n'$-complex symmetric with a conjugation $D$‎, ‎then $T \otimes S$ is $(m+m'-1,n+n'-1,C \otimes D)$-isosymmetric‎.