Publications

⭐  Estimates and Higher-Order Spectral Shift Measures in Several Variables-  (with Arup Chattopadhyay and Saikat Giri), Linear Algebra and its Applications 698 (2024), 102--134.

          Summary: We establish estimates for traces of higher-order derivatives of multivariate operator functions, extending previous results to include higher-order spectral shift measures for tuples of commuting contractions under Hilbert-Schmidt perturbations. 

⭐  Second-order trace formulae - (with Arup Chattopadhyay and Soma Das), Mathematische Nachrichten (2024), 28 pp,  https://doi.org/10.1002/mana.202200295.

   Summary: This study presents a new proof of Koplienko's trace formula for pairs of contractions, specifically when the initial operator $T_0$ is normal. We use a linear path and employ a finite-dimensional approach, inspired by Voiculescu's method in Krein's trace formula. This extension of the Koplienko trace formula to a class of contraction pairs involves the Schäffer matrix unitary dilation. Additionally, we derive the trace formula for pairs of self-adjoint and maximal dissipative operators through the Cayley transform. Finally, we extend the Koplienko-Neidhardt trace formula for a specific class of contraction pairs via multiplicative paths, using finite-dimensional approximation. 

⭐  Schmidt subspaces of block Hankel operators - (with Arup Chattopadhyay and Soma Das), to appear in the Journal of Operator Theory (2024), 23 pp.

   Summary: This article establishes a connection between Schmidt subspaces of bounded Hankel operators in vector-valued Hardy spaces and nearly $S^*$-invariant subspaces. We demonstrate that these subspaces possess finite defect in general, offering a concise alternative proof for the characterization of Schmidt subspaces in scalar-valued Hardy space. Our findings complement the work of G'{e}rard and Pushnitski, enhancing understanding of Schmidt subspace structures. 

⭐ Approximation of the spectral action functional in the case of $\tau$-compact resolvents - (with Arup Chattopadhyay, Anna Skripka), Integral Equations Operator Theory 95 (2023), Paper No. 20, 15 pp.

 Summary: We derive estimates and representations for Taylor approximation remainders of the spectral action functional $V\mapsto\tau(f(H_0+V))$ on bounded self-adjoint perturbations. Our results extend those in [A.~Skripka: Taylor asymptotics of spectral action functionals. J. Operator Theory, 80  (2018), no. 1, 113--124], accommodating a broader class of functions and stronger estimates when the resolvent of $H_0$ belongs to the noncommutative $L^n$-space. 

⭐  On Isometric Embeddability of $S_q^m$ into $S_p^n$ as  Quasi-Banach spaces- (with Arup Chattopadhyay, Guixiang Hong, and Samya K. Ray )  Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, (in press) (2023), DOI: 10.1017/prm.2023.54, 24pp.

  Summary: This article broadens our earlier study, progressing from the Banach setting to the realm of Quasi-Banach spaces.

⭐  Higher-order spectral shift for pairs of contractions via multiplicative path- (with Arup Chattopadhyay), New York Journal of Mathematics 29 (2023), 25pp.

 Summary: In a previous study, Marcantognini and Morán derived the Koplienko-Neidhardt trace formula for pairs of contractions and maximal dissipative operators using a multiplicative path. This article builds on their work, proving the existence of higher-order spectral shift functions for these pairs through a similar approach.

⭐  Characterization of C-symmetric Toeplitz operators for a class of conjugations in Hardy spaces - (with Arup Chattopadhyay, Soma Das, and Srijan Sarkar), Linear and Multilinear Algebra, 71 (2023), no. 12, 2026-2048.

 Summary: This article introduces a novel set of conjugations in the scalar-valued Hardy space and characterizes complex symmetric Toeplitz operators $T_{\phi}$ with respect to these conjugations in different scenarios. Additionally, it provides a characterization of a complex symmetric block Toeplitz operator $T_{\Phi}$ on the vector-valued Hardy space using specific conjugations from previous works. 

⭐  Isometric embeddability of $S_q^m$ into $S_p^n$ - (with Arup Chattopadhyay, Guixiang Hong, Avijit Pal, and Samya Kumar Ray ) Journal of Functional Analysis 282 (2022), 29 pp.

 Summary: This paper investigates the isometric embedding of the Schatten class $S_q$ into the Schatten class $S_p$ ($1\leq p\neq q\leq \infty$). It concludes that, in most instances, the sole feasible scenario is the isometric embedding of $S_2$ into $S_p$. Our work complements the work of Junge, Parcet, Xu, and others in non-commutative $L_p$-space embedding theory. We employ novel elements from perturbation theory, including the Kato-Rellich theorem, multiple operator integrals, and Birkhoff-James orthogonality, along with meticulous case analysis. 

⭐  The Koplienko-Neidhardt trace formula for unitaries—a new proof - (with Arup Chattopadhyay and Soma Das)  Journal of Mathematical Analysis and Applications 505 (2022), no. 1, Paper No. 125467, 21 pp.

  Summary: Koplienko established a trace formula for perturbations of self-adjoint operators by Hilbert-Schmidt class operators. Neidhardt later extended this to unitary operators in 1988. This article offers an alternative proof of the Koplienko-Neidhardt trace formula for unitary operators by reducing the problem to a finite-dimensional one, similar to Voiculescu's proof of Krein's trace formula.

⭐  Kernels of perturbed Toeplitz operators in vector-valued Hardy spaces - (with Arup Chattopadhyay and Soma Das)  Advances in Operator Theory 6 (2021), no. 3, Paper No. 49, 28 pp.

  Summary: Liang and Partington recently demonstrated that kernels of perturbed Toeplitz operators exhibit near invariance under the backward shift operator in scalar-valued Hardy space. This article extends their result to a vectorial context, revealing kernels of perturbed Toeplitz operator in terms of backward shift-invariant subspaces, using relevant recent theorems. 

⭐  Almost invariant subspaces of the shift operator on vector-valued Hardy spaces - (with Arup Chattopadhyay and Soma Das) Integral Equations Operator Theory 92 (2020), no. 6, Paper No. 52, 15 pp. 

  Summary: The article investigates nearly invariant subspaces of finite defect for the backward shift operator in the vector-valued Hardy space. It extends a previous result by Chalendar, Gallardo, and Partington and provides a comprehensive description of almost invariant subspaces for both the shift and its adjoint within the same space.