arXiv subjects

S. H. Kulkarni

Publications and source records attributed to S. H. Kulkarni.

6 recordsLinked to original sources

arXiv2022

Spectral representation of absolutely minimum attaining unbounded normal operators

Let $T:D(T)\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every closed subspace $M$ of $H_1$, the restriction operator $T|_M:D(T)\cap M\rightarrow H_2$ attains its minimum modulus $m(T|_{M})$. That is, there exists $x \in D(T)\cap M$ with $\|x\|= 1$ and $\|T(x)\| = \inf \{\|T(m)\|: m \in D(T) \cap M: \|m\|=1\}$. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent.

math.FA↗

arXiv2021

$G_1$ class elements in a Banach algebra

Let $A$ be a complex unital Banach algebra with unit $1$. An element $a\in A$ is said to be of \textit{$G_{1}$-class} if $$\|(z-a)^{-1}\|=\frac{1}{\text{d}(z,σ(a))} \quad \forall z\in \mathbb{C}\setminus σ(a).$$ Here $d(z, σ(a))$ denotes the distance between $z$ and the spectrum $σ(a)$ of $a$. Some examples of such elements are given and also some properties are proved. It is shown that a $G_1$-class element is a scalar multiple of the unit $1$ if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if $T$ is a $G_1$ class operator on a Banach space $X$, then every isolated point of $σ(T)$ is an eigenvalue of $T$. If, in addition, $σ(T)$ is finite, then $X$ is a direct sum of eigenspaces of $T$.

math.FA↗

arXiv2019

Absolutely minimum attaining closed operators

We define and discuss properties of the class of unbounded operators which attain minimum modulus. We establish a relationship between this class and the class of norm attaining bounded operators and compare the properties of both. Also we define absolutely minimum attaining operators (possibly unbounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of them is that every such operator has a non trivial hyperinvariant subspace.

math.FA↗

arXiv2018

Operators that attain the reduced minimum

Let $H_1, H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator from its domain $D(T)$, a dense subspace of $H_1$, into $H_2$. Let $N(T)$ denote the null space of $T$ and $R(T)$ denote the range of $T$. Recall that $C(T) := D(T) \cap N(T)^{\perp}$ is called the {\it carrier space of} $T$ and the {\it reduced minimum modulus } $γ(T)$ of $T$ is defined as: $$ γ(T) := \inf \{\|T(x)\| : x \in C(T), \|x\| = 1 \} .$$ Further, we say that $T$ {\it attains its reduced minimum modulus} if there exists $x_0 \in C(T) $ such that $\|x_0\| = 1$ and $\|T(x_0)\| = γ(T)$. We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved.

math.FA↗

arXiv2016

On the denseness of minimum attaining operators

Let $H_1,H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator (not necessarily bounded). It is proved that for each $ε>0$, there exists a bounded operator $S$ with $\|S\|\leq ε$ such that $T+S$ is minimum attaining. Further, if $T$ is bounded below, then $S$ can be chosen to be rank one.

math.FA↗

arXiv2015

Perturbation of closed range operators and Moore-Penrose inverse

Let $H_1,H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subseteq H_1$ and $T^{\dagger}$ be the Moore-Penrose inverse of $T$. Let $S:H_1\rightarrow H_2$ be a bounded operator. In this article we focus our attention on the following questions: $1.$ Under what conditions closedness of range of $T$ will imply the closedness of range of $T+S$? $2.$ What is the relation between $T^{\dagger}$ and $(T+S)^{\dagger}$? $3.$ What is the relation between $T^{\dagger}$ and $S^{\dagger}$?.

math.FA↗