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1
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An inequality for spherical Cauchy dual tuples

100%
Chavan S.
Colloquium Mathematicum
|
2013
|
tom 131
|
nr 2
265-271
EN
Let T be a spherical 2-expansive m-tuple and let $T^{𝔰}$ denote its spherical Cauchy dual. If $T^{𝔰}$ is commuting then the inequality $∑_{|β|=k} (β!)^{-1}(T^{𝔰})^{β}(T^{𝔰})*^{β} ≤ (k+m-1 \atop k) ∑_{|β|=k} (β!)^{-1}(T^{𝔰})*^{β}(T^{𝔰})^{β}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.
2
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Co-analytic, right-invertible operators are supercyclic

100%
Chavan S.
Colloquium Mathematicum
|
2010
|
tom 119
|
nr 1
137-142
EN
Let 𝓗 denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space 𝓗, let 𝓑(𝓗) denote the algebra of bounded linear operators on 𝓗. We show that for any co-analytic, right-invertible T in 𝓑(𝓗), αT is hypercyclic for every complex α with $|α| > β^{-1}$, where $β ≡ inf_{||x||=1}||T*x|| > 0$. In particular, every co-analytic, right-invertible T in 𝓑(𝓗) is supercyclic.
3
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On operators close to isometries

100%
Chavan S.
Studia Mathematica
|
2008
|
tom 186
|
nr 3
275-293
EN
We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p-isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators.
4
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On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

100%
Chavan S.
Studia Mathematica
|
2011
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tom 203
|
nr 2
129-162
EN
The Cauchy dual operator T', given by $T(T*T)^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T' admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below). This connection, which is new even in the bounded case, enables us to establish some interesting properties of unbounded 2-hyperexpansions and their Cauchy dual operators such as the completeness of eigenvectors, the hypercyclicity of scalar multiples, and the wandering subspace property. In particular, certain cyclic 2-hyperexpansions can be modelled as the forward shift ℱ in a reproducing kernel Hilbert space of analytic functions, where the complex polynomials form a core for ℱ. However, unlike unbounded subnormals, $(T*T)^{-1}$ is never compact for unbounded 2-hyperexpansive T. It turns out that the spectral theory of unbounded 2-hyperexpansions is not as satisfactory as that of unbounded subnormal operators.
5
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Completely monotone functions of finite order and Agler's conditions

64%
Chavan S., Sholapurkar V.
Studia Mathematica
|
2015
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tom 226
|
nr 3
229-258
EN
Motivated by some structural properties of Drury-Arveson d-shift, we investigate a class of functions consisting of polynomials and completely monotone functions defined on the semi-group ℕ of non-negative integers, and its operator-theoretic counterpart which we refer to as the class of completely hypercontractive tuples of finite order. We obtain a Lévy-Khinchin type integral representation for the spherical generating tuples associated with such operator tuples and discuss its applications.
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