On class A operators
We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive. 1. Introduction. LetH be a complex separable Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. If T 2 L(H), we write (T ), ap(T ), and e(T ) for the spectrum, the approxi- mate point spectrum, and the essential spectrum, respectively, and write r(T ) = supfj j : 2 (T )g for the spectral radius of T. An operator T 2 L(H) is said to be p-hyponormal if (TT ) p (T T ) p , where 0 < p <1. In particular, 1-hyponormal operators and 1 -hyponormal operators are called hyponormal operators and semi-hyponormal operators, respectively. An arbitrary operator T 2L(H) has a unique polar decomposition T = UjTj, wherejTj = (T T ) 1=2 and U is a partial isometry satisfying kerU = kerjTj = kerT and kerU = kerT . Associated with T is the operator jTj 1=2 UjTj 1=2 called the Aluthge transform of T , and denoted throughout this paper by b