Similarity to an isometry of a composition operator

We study the composition operators which are similar to an isometry on the classical Hardy space .

     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.     

Real operator algebras and real completely isometric theory

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485–496, 2003, Illinois J Math 47(4):1047–1062, 2003), of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, $C^*$ -envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided $M$ -ideal theory and characterize one-sided $M$ -ideals in real $C^*$ -algebras and real operator algebras with contractive approximate identity.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

A formula for the generalized resolvent of an isometric operator for the exterior of the unit disk

In this note we establish the relationship between characteristic functions and generalized resolvents of isometric operators for the exterior of the unit disk. A certain sufficient condition is given in terms of characteristic functions that the form of the formula for the generalized resolvent be preserved across the unit circle.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 849–856, June, 1973.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L₀ be a positive definite closed linear operator with domain of definition D(L₀) dense in the Hilbert space H; let(, ₁, ₂) be the positive boundary value space of the operator L₀ such that the restriction of L ₀ ^* to ker ₂ is the Friedrichs extension of the operator L₀. We establish a test for nonnegativity of an operator T of the form Ty=L ₀ ^* y+^*(₁–C)y, y D(T)= ker(₂+), where :H and C: are respectively a compact operator and a bounded nonnegative operator.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 30–33.     

Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on p-forms

In this paper, we study the first eigenvalue of the Dirichlet-to-Neumann operator acting on differential forms of a Riemannian manifold with boundary isometrically immersed in some Euclidean space. We give a lower bound of the integral energy of p-forms in terms of its first eigenvalue associated with $(p?1)$-forms. We also find a lower bound for the gap between two consecutive first eigenvalues in terms of the curvature of the boundary.     

On an integral operator

In this paper we derive some criteria for univalence of a general integral operator for analytic functions in the open unit disk.     

On an operator inequality

Jensen's operator inequality characterizes operator convex functions of two variables (F. Hansen, Proc. Amer. Math. Soc. 125 (1997) 2093–2102). We give a simplified proof of this theorem formulated for matrices.     

Convergence of an operator series

The paper gives a necessary and sufficient condition on the spectrum of a bounded linear operator on Banach space for the convergence of the series ₀ T(I-T ²) ⁿ . Some properties of the sum are investigated.