Geodesic
The Bishop's property (β) for class A operators
Filomat, Tome 37 (2023) no. 23, p. 7807

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DOI

We say that an operator T on a Hilbert space H has the Bishop's property (β) if for an arbitrary open set U ⊂ C and analytic functions f n : U → H with ∥(T − z) f n (z)∥ converges to 0 uniformly on every compact subset of U as n → ∞ then ∥ f n ∥ converges to 0 uniformly on every compact subset of U as n → ∞. An operator T on H is called to be hyponormal if T * T ≥ TT *, and T is called to be class A if T * T ≤ |T 2 |. In this papaer, we give an elementary proof of the assertion that every hyponormal operator has the Bishop's property (β). And we show that every class A operator has the Bishop's property (β). Moreover, we also show a class A operator T is similar to a hyponormal operator if T is invertible, and hence T has the growth condition (G 1).
DOI : 10.2298/FIL2323807U
Classification : 47A10, 47B20
Keywords: hyponormal, p-hyponormall operator, w-hyponormal operator, class A, Bishop’s property 
Atsushi Uchiyama; Kotaro Tanahashi. The Bishop's property (β) for class A operators. Filomat, Tome 37 (2023) no. 23, p. 7807 . doi: 10.2298/FIL2323807U
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     title = {The {Bishop's} property (\ensuremath{\beta}) for class {A} operators},
     journal = {Filomat},
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     year = {2023},
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     doi = {10.2298/FIL2323807U},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2323807U/}
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