2025/05/11 by Priyanka Grover, Grover, Priyanka, Krishna Kumar Gupta +3
Mathematics · #15A60 #46B20 #46G05 #47A30 #47L10 #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2505.06925
openalex publication_date 2025/05/11 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
We present an expression for the right hand derivative of the B(H,K) norm generalizing the result for K=H in [D. J. Ke\checkcki\gravec, Gateaux derivative of B(H) norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the B(H, K) norm. For tuples of operators A,X∈ B(H, Hd), we give a characterization for \boldsymbol 0 to be a best approximation to the subspace \mathbb Cd X, generalizing a similar result for \mathbb Cd I in [P. Grover, S. Singla, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of ε-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for A∈ B(H,K) to be ε-Birkhoff orthogonal to a subspace of B(H,K), when A is compact.