ACTA issues

Spectrum of class $p$-$wA(s,t)$ operators

Muneo Chō, M. H. M. Rashid, Kotaro Tanahashi, Atsushi Uchiyama

Acta Sci. Math. (Szeged) 82:3-4(2016), 641-649

Abstract. Let $T=U|T|$ be the polar decomposition of a bounded linear operator on a complex Hilbert space. $T $ is called a class $p$-$wA(s,t)$ operator if $(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{tp}{s+t}}\geq |T^{*}|^{2tp}$ and $(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{sp}{s+t}}\leq |T|^{2sp}$ where $0 < s, t $ and $0 < p \leq1$. We investigate spectral properties of a class $p$-$wA(s,t)$ operator $T$. We prove that if $s + t = 1$ and $\lambda\not = 0 $ is an isolated point of the spectrum $\sigma(T)$ then the Riesz idempotent $E$ with respect to $\lambda $ is self-adjoint and $ {\rm ran } E = \ker(T- \lambda ) = \ker((T-\lambda )^{*})$. Also, we prove relating results.

DOI: 10.14232/actasm-015-275-0

AMS Subject Classification (1991): 47B20

Keyword(s): class $A$ operator, class $p$-$wA(s, t)$ operator, Riesz idempotent

Received March 27, 2015, and in revised form May 6, 2015. (Registered under 25/2015.)