On the spectrum of n-tuples of p-hyponormal operators

Duggal, B. P.

Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 123-131

Résumé

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let HU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. HU(l/2) operators were introduced by Xia and HU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); HU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, HU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε HU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the HW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.

Détail

DOI : 10.1017/S0017089500032419

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