ACTA issues

$p$-hyponormal operators and invariant subspaces

B. P. Duggal

Acta Sci. Math. (Szeged) 64:1-2(1998), 249-257

Abstract. Given a $p$-hyponormal operator $(0< p< 1/2)$ $T$ on a Hilbert space, define operators $\hat T$ and $\tilde T$ by $\hat T=| T| ^{1/2}U| T| ^{1/2}$ and $\tilde T=| \hat T| ^{1/2}V| \hat T| ^{1/2}$, where the partial isometries $U$ and $V$ are as in the polar decompositions $T=U| T| $ and $\hat T=V| \hat T| $. The operator $\tilde T$ is then hyponormal. We show that $T$ has a non-trivial invariant subspace if and only if $\tilde T$ does, and that if $T$ does not have a non-trivial invariant subspace, then $T$ is the compact perturbation of a normal operator. We also consider upper triangular operators with $p$-hyponormal entries along the main diagonal.

AMS Subject Classification (1991): 47A15, 47B20

Keyword(s): p, -hyponormal operator, invariant subspace, compact perturbation

Received May 6, 1997. (Registered under 2660/2009.)