Algebraic extensions of semi-hyponormal operators

In this paper, we show that algebraic extensions of semi-hyponormal operators (defined below) are subscalar. As corollaries we get the following:

(1)	Everyk-quasihyponormal operator is subscalar.
(2)	Every algebraic extension of Aluthge transforms ofp-hyponormal operators is subscalar.

From these results and Es] it is known that such operators with rich spectra have nontrivial invariant subspaces.The second author was supported by the grant for the promotion of scientic research in women's universities.