Multipliers for operator-valued Bessel sequences, generalized Hilbert-Schmidt and trace classes

2019/08/29 by Krishna, K. Mahesh, Johnson, P. Sam, Mohapatra, R. N.
#42C15 #46C05 #47A05 #47L20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1908.11059

Abstract

Let \λ_n\_n ∈ ℓ^∞(ℕ). In 1960, R. Schatten \citeSCHATTEN studied operators of the form ∑_n=1^∞λ_n (x_n⊗ y_n), where \x_n\_n, \y_n\_n are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \citeBALAZS3 generalized this by replacing \x_n\_n and \y_n\_n by Bessel sequences. In this paper, we generalize this by studying the operators of the form ∑_n=1^∞λ_n (A^*_nx_n⊗ B^*_ny_n), where \A_n\_n and \B_n\_n are operator-valued Bessel sequences and \x_n\_n, \y_n\_n are sequences in the Hilbert space such that \‖x_n‖‖y_n‖\_n ∈ ℓ^∞(ℕ). We next generalize the classes of Hilbert-Schmidt and trace class operators.

Related

• Multipliers on Hilbert Spaces of Dirichlet Series
• Generalized Bessel multipliers in Hilbert spaces
• Multipliers for Lipschitz p-Bessel sequences in metric spaces
• Sharp multiplier theorem for multidimensional Bessel operators
• Unconditionally convergent multipliers and Bessel sequences