Inclusion Relations for Some Lp-Spaces of Operator-Valued Functions
✍ Scribed by L. Klotz
- Publisher
- John Wiley and Sons
- Year
- 1991
- Tongue
- English
- Weight
- 389 KB
- Volume
- 150
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Let (Q, a, p) be a positive measure space and D ( p ) , 0 < p 5 00, the space of (equivalence classes of) functions which are p-integrable with respect to p. In a nice short paper [9] A. VILLANI stated necessary and sufficient conditions for the measure p such that inclusions of the form D ( p ) D ( p ) hold. In [5] spaces of (equivalence classes of) operator-valued functions being y-integrable with respect to a certain operator-valued measure were introduced. It was proved that some elementary facts concerning the spaces D ( p ) remain true for these spaces of operator-valued functions. The present paper deals with the generalization of VILLANI'S results to these spaces. The logical structure of the proofs is the same as in VILLANI'S paper. However, one meets with additional technical difficulties, because the appearance of unbounded operators leads to different concepts of equivalence for different values of p.
Throughout the paper, we use the following notations. For a linear operator X , A ( X ) denotes its range. For a bounded linear operator X , 1x1 and X* denote its operator norm and the generalized inverse, respectively. The symbol s-lim stands for the limit with respect to the strong operator topology of a sequence of operators and l i m for the upper limit of a sequence of numbers. Finally, by N we denote the set of positive integerg and by C the set of complex numbers.
Let H and K be two separable HILBERT spaces over C, whose dimensions are dim H 5 w and dim K 5 00. Since no confusion may occur, we denote the scalar product and the norm in both spaces H and K by (., -) and 11. 11, respectively. Let 3 (urn, resp.) be the BANACII space of all bounded (compact, resp.) linear operators of H into K and 3 ( H ) the BANACH algebra of all bounded linear operators in H .