Characterization of unbounded spectral operators with spectrum in a half-line |
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Authors: | Shmuel Kantorovitz |
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Institution: | 1. Temple University, 19122, Philadelphia, Pennsylvania, USA 2. Bar-Ilan University, Ramat-Gan, Israel
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Abstract: | LetT be a possibly unbounded linear operator in the Banach spaceX such thatR(t)=(t+T)?1 is defined onR +. LetS=TR(I?TR) and letB(.,.) denote the Beta function. Theorem 1.1.T is a scalar-type spectral operator with spectrum in 0, ∞) if and only if $$sup\left\{ {B\left( {k,k} \right)^{ - 1} \int_0^\infty {\left| {x*S^k \left( t \right)x} \right|{{dt} \mathord{\left/ {\vphantom {{dt} t}} \right. \kern-\nulldelimiterspace} t};\left\| x \right\| \leqslant 1,} \left\| {x*} \right\| \leqslant 1,k \geqslant 1} \right\}< \infty .$$ A “local” version of this result is formulated in Theorem 2.2. |
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