A PUTNAM AREA INEQUALITY FOR THE SPECTRUM OF n-TUPLES OF p-HYPONORMAL OPERATORS

We prove an n-tuple analogue of the Putnam area inequality for the spectrum of a single p-hyponormal operator. Let B…H† denote the algebra of operators (i.e. bounded linear transformations) on a separable Hilbert space H. The operator A 2 B…H† is said to be p-hyponormal, 0 < p 1, if jA j2p jAj2p. Let H…p† denote the class of p-hyponormal operators. Then H…1† consists of the class of p-hyponormal operators and H…2† consists of the class of semi-hyponormal operators introduced by D. Xia. (See [11, p. 238] for the appropriate reference.) H…p† operators for a general p with 0 < p < 1 have been studied by a number of authors in the recent past; (see [3, 4, 5] for further references). Generally speaking, H…p† operators (0 < p < 1) have spectral properties very similar to those of hyponormal operators. In particular, a Putnam inequality relating the norm of the commutator Dp ˆ jAj2p ÿ jA j2p of A 2 H…p† to the area of the spectrum …A† of A holds; indeed jjDpjj p …