Compact Operators Under Orlicz Function

Ma Zhenhua (), Ji Kui () and Li Yucheng ()
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Ma Zhenhua: Hebei Normal University
Ji Kui: Hebei Normal University
Li Yucheng: Hebei Normal University

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 4, 1633-1649

Abstract: 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 $${T_{1 — {{\left| z \right|}^2}}}$$ T 1 − ∣ z ∣ 2 on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ L α 2 ( D ) belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ 1 = ∑ n ≥ 0 φ ( 1 ( n + 2 ) ‖ T 1 − ∣ z ∣ 2 ‖ φ ) . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ ‖ T 1 − ∣ z ∣ 2 ‖ p = [ ∑ n ≥ 0 1 ( n + 2 ) p ] 1 p = ( ζ ( p ) − 1 ) 1 p , where ζ(p) is the Riemann function.

Keywords: Noncommutative Orlicz sequence spaces; compact operators; Schatten classes; Orlicz function; 2010; 46L89; 47L25 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13226-020-0486-5

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