ACTA issues

Spectrum of class $A(s,t)$ operators

Atsushi Uchiyama, Kôtarô Tanahashi, Jun Ik Lee

Acta Sci. Math. (Szeged) 70:1-2(2004), 279-287

Abstract. Let $T = U|T|$ be the polar decomposition of a bounded linear operator $T$ on a complex Hilbert space ${\cal H}$ and $T(s,t)=|T|^sU|T|^t$ for $ 0 < s, t \leq1$. $T$ is called a class $A(s,t)$ operator if $ |T(s,t)|^{2t\over s+t} \geq |T|^{2t} $, which is a further generalization of $p$-hyponormal or $\log $-hyponormal operator. We shall show that if $T$ is a class $A(s,t)$ operator for $ 0 < s, t\leq1$, then (i) $ \sigma(T(s,t)) = \{r^{s+t}e^{i\theta } : re^{i\theta } \in\sigma (T)\} $, (ii) for each non-zero isolated point $\lambda =re^{i\theta }$ of $\sigma(T)$, the Riesz idempotent $E$ for $T$ with respect to $\lambda $ is self-adjoint and coincides with the Riesz idempotent $E(s,t)$ for $T(s,t)$ with respect to $\lambda_{s+t}=r^{s+t}e^{i\theta }$.

AMS Subject Classification (1991): 47A10, 47B20

Keyword(s): A, classoperator, A(s, class, operator, t), Riesz idempotent

Received September 30, 2003. (Registered under 5813/2009.)