Abstract. In this paper we prove that if $T\in B({\mathcal H})$ is a pure class $p$-$wA(s,t)$ operator ($0 < s, t, s + t =1$ and $0 < p \leq 1$) with dense range such that $0\notin \sigma _{p}(T)$, then $T$ has a non-trivial invariant subspace if and only if its second generalized Aluthge transformation $\tilde {T}(s,t)$ has a non-trivial invariant subspace. Further, we study some conditions for class $p$-$wA(s,t)$ operators to have a non-trivial invariant subspace.

DOI: 10.14232/actasm-020-775-8

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

Keyword(s): $p$-hyponormal operator, class $p$-$wA(s, t)$ operator, invariant subspaces

received 25.5.2020, revised 25.8.2020, accepted 6.9.2020. (Registered under 525/2020.)