Showing papers by "Muneo Chō published in 2018"

On symmetric and skew-symmetric operators

Abstract: In this paper we show many spectral properties that are inherited by m-complex symmetric and m-skew complex symmetric operators and give new results or recapture some known ones for complex symmetric operators.

Spectral properties of n-normal operators

Abstract: For a bounded linear operator T on a complex Hilbert space and n ϵ N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies σ(T) ∩ (-σ(T))  {0}, then T is isoloid and σ(T) = σa(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z) ker(T-w). And if non-zero number z ϵ C is an isolated point of σ(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying σ(T) ∩(-σ(T)) = 0, then Weyl’s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

On $$\varvec{(m,C)}$$ ( m , C ) -Isometric Commuting Tuples of Operators on a Hilbert Space

Abstract: Inspired by recent works on (m, C)-isometric and [m, C]-isometric operators on Hilbert spaces studied respectively in Chō et al. (Complex Anal. Oper. Theory 10:1679–1694, 2016; Filomat 31:7, 2017), in this paper we introduce the class of (m, C)-isometries for tuple of commuting operators. This is a generalization of the class of (m, C)-isometric operators. A commuting tuples of operators $$\mathbf{\large T}=(T_1,\ldots ,T_d)\in {\mathcal {B}}^{(d)}({\mathcal {H}})$$ is said to be (m, C)-isometric tuple if $$\begin{aligned} {{\mathcal {Q}}}_{m}(\mathbf{T}):=\sum _{0\le k\le m}(-1)^{m-k}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( \sum _{|\beta |=k}\frac{k!}{\beta !}\mathbf{\large T}^{*\beta }C\mathbf{\large T}^{\beta }C\right) =0 \end{aligned}$$ for some positive integer m and some conjugation C. We consider a multivariable generalization of these single variable (m, C)-isometric operators and explore some of their basic properties.

Complex isosymmetric operators

Abstract: 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.