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Compact Operators Under Orlicz Function
- Source :
- Indian Journal of Pure and Applied Mathematics. 51:1633-1649
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- In this paper we investigate the compact operators under Orlicz function, named noncommutative Orlicz sequence space (denoted by Sϕ(ℌ)), where ℌ is a complex, separable Hilbert space. We will show that the space generalizes the Schatten classes Sp(ℌ) and the classical Orlicz sequence space respectively. After getting some relations of trace and norm, we will give some operator inequalities, such as Holder inequality and some other classical operator inequalities. Also we will give the dual space and reflexivity of Sϕ(ℌ) which generalizes the results of Sϕ(ℌ). Finally, as an application, we will show that the Toeplitz operator $${T_{1 — {{\left| z \right|}^2}}}$$ on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ , where ζ(p) is the Riemann function.
- Subjects :
- Hölder's inequality
Dual space
Applied Mathematics
General Mathematics
010102 general mathematics
Compact operator
01 natural sciences
Noncommutative geometry
Sequence space
Combinatorics
Riemann hypothesis
symbols.namesake
Bergman space
0103 physical sciences
symbols
010307 mathematical physics
0101 mathematics
Mathematics
Toeplitz operator
Subjects
Details
- ISSN :
- 09757465 and 00195588
- Volume :
- 51
- Database :
- OpenAIRE
- Journal :
- Indian Journal of Pure and Applied Mathematics
- Accession number :
- edsair.doi...........098b75099821646813a642e99dd523b4