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Soma Das

Publications and source records attributed to Soma Das.

15 recordsLinked to original sources
arXiv2026

Foguel-type operators similar to contractions

Pisier's celebrated counterexample to Halmos's similarity-to-contractions problem was based on $2 \times 2$ upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz operators, one can formulate another $2^3 -1 = 7$ types of $2 \times 2$ upper triangular block operator matrices, which we refer to as Foguel-type operators. In this paper, we give a complete characterization of all the seven Foguel-type operators being similar to contractions.

math.FA↗
arXiv2024

Invariant subspaces of perturbed backward shift

We represent closed subspaces of the Hardy space that are invariant under finite-rank perturbations of the backward shift. We apply this to classify almost invariant subspaces of the backward shift and represent a more refined version of nearly invariant subspaces. Kernels of certain perturbed Toeplitz operators are examples of the newly introduced nearly invariant subspaces.

math.FA↗
arXiv2024

Paired and Toeplitz + Hankel operators

We present complete classifications of Toeplitz + Hankel operators on vector-valued Hardy spaces and classify paired operators on $L^2(\mathbb{T})$. We also study the latter class through the lens of inner functions on the disc.

math.FA↗
arXiv2024

Second order trace formulae

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later, Neidhardt introduced a similar formula in the case of pair of unitaries $(U,U_0)$ via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev \cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new proof of the Koplienko trace formula in the case of pair of contractions $(T,T_0)$, where the initial operator $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko-Neidhardt trace formula for a class of pair of contractions $(T,T_0)$ via multiplicative path.

math.FA↗
arXiv2023

Schmidt subspaces of block Hankel operators

In scalar-valued Hardy space, the class of Schmidt subspaces for a bounded Hankel operator are closely related to nearly $S^*$-invariant subspaces, as described by Gérard and Pushnitski. In this article, we prove that these subspaces in the context of vector-valued Hardy spaces are nearly $S^*$-invariant with finite defect in general. As a consequence, we obtain a short proof of the characterization results concerning the Schmidt subspaces in scalar-valued Hardy space in an alternative way. Thus, our work complements the work of Gérard and Pushnitski regarding the structure of Schmidt subspaces.

math.FA↗
arXiv2023

Commutants and Complex Symmetry of Finite Blaschke Product Multiplication Operator in $\bm{L^2(\T)}$

Consider the multiplication operator $M_{B}$ in $L^2(\T)$, where the symbol $B$ is a finite Blaschke product. In this article, we characterize the commutant of $M_{B}$ in $L^2(\T)$, noting the fact that $L^2(\T)$ is not an RKHS. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\T)$. Moreover, we analyze their properties while keeping the whole Hardy space, model space, and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke.

math.FA↗
arXiv2021

Characterization of C-Symmetric Toeplitz operators for a Class of Conjugations in Hardy Spaces

In this article, we introduce a new class of conjugations in the scalar valued Hardy space $H^2_{\mathbb{C}}(\mathbb{D})$ and provide a characterization of a complex symmetric Toeplitz operator $T_ϕ$ with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator $T_Φ$ on the vector valued Hardy space $\hdct$ with respect to certain conjugations introduced in \cite{CamaGP,KangKoLee,LeeKo2019}.

math.FA↗
arXiv2020

Koplienko-Neidhardt trace formula for unitaries $-$ A new proof

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later in 1988, a similar formula was obtained by Neidhardt \cite{NH} in the case of unitary operators. In this article, we give a still another proof of Koplienko-Neidhardt trace formula in the case of unitary operators by reducing the problem to a finite dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}.

math.FA↗
arXiv2020

Study of nearly invariant subspaces with finite defect in Hilbert spaces

In this article, we briefly describe nearly $T^{-1}$ invariant subspaces with finite defect for a shift operator $T$ having finite multiplicity acting on a separable Hilbert space $\mathcal{H}$ as a generalization of nearly $T^{-1}$ invariant subspaces introduced by Liang and Partington in \cite{YP}. In other words we characterize nearly $T^{-1}$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in \cite{CDP}. Furthermore, we also provide a concrete representation of the nearly $T_B^{-1}$ invariant subspaces with finite defect in a scale of Dirichlet-type spaces $\mathcal{D}_α$ for $α\in [-1,1]$ corresponding to any finite Blashcke product $B$.

math.FA↗
arXiv2020

Almost invariant subspaces of the shift operator on vector-valued Hardy spaces

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

math.FA↗
arXiv2020

Kernels of Perturbed Toeplitz Operators in vector-valued Hardy spaces

Recently, Liang and Partington \cite{YP} show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article we provide a vectorial generalization of a result of Liang and Partington. As an immediate application we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (\cite{CDP, OR}) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.

math.FA↗
arXiv2008

Effect of intrinsic instability of cantilevers on static mode Atomic Force Spectroscopy

We show that the static force spectroscopy curve is significantly modified due to presence of intrinsic cantilever instability. This instability acts in tandem with such instabilities like water bridge or molecular bond rupture and makes the static force spectroscopy curve (including "jump-off-contact") dependent on the step-size of the movement of sample stage. A model has been proposed to explain the data. This has been further validated by applying an electric field between tip and substrate which modifies the tip-substrate interaction.

cond-mat.mtrl-sci↗
arXiv2008

A method to quantitatively evaluate Hamaker constant using the jump-into-contact effect in Atomic Force microscopy

We find that the jump-into-contact of the cantilever in the atomic force microscope (AFM) is caused by an inherent instability in the motion of the AFM cantilever. The analysis is based on a simple model of the cantilever moving in a nonlinear force field. We show that the jump-into-contact distance can be used to find the interaction of the cantilever tip with the surface. In the specific context of the attractive van der Waals interaction, this method can be realized as a new method of measuring the Hamaker constant for materials. The Hamaker constant is determined from the deflection of the cantilever at the jump-into-contact using the force constant of the cantilever and the tip radius of curvature, all of which can be obtained by measurements. The results have been verified experimentally on a sample of cleaved mica, a sample of Si wafer with natural oxide and a silver film, using a number of cantilevers with different spring constants. We emphasize that the method described here is applicable only to surfaces that have van der Waals interaction as the tip-sample interaction. We also find that the tip to sample separation at the jump-into-contact is simply related to the cantilever deflection at this point, and this provides a method to exactly locate the surface.

cond-mat.mtrl-sci↗