537 research outputs found
The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality
The validity of the von-Neumann inequality for commuting - tuples of
matrices remains open for . We give a partial answer to
this question, which is used to obtain a necessary condition for the
Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc
In the special case of (which follows from Ando's theorem as well), this
necessary condition is made explicit. An alternative approach to the
Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of
adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence,
a class of polynomials are isolated for which a complete solution to the
Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural
generalization of the Hankel operators on the Hardy space of
then becomes apparent. Many of our results remain valid for any however, the computations are somewhat cumbersome for and are
omitted. The inequality , where
is the complex Grothendieck constant and
is due to Varopoulos. Here the
supremum is taken over all complex polynomials in variables of degree
at most and commuting - tuples of
contractions. We show that obtaining a slight improvement in the inequality of Varopoulos.
We show that the normed linear space has no isometric
embedding into complex matrices for any and
discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore
on 20th July, 201
Linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance without landesman-lazer conditions
We are concerned with the solvability of linear second order elliptic partial
differential equations with nonlinear boundary conditions at resonance, in
which the nonlinear boundary conditions perturbation is not necessarily
required to satisfy Landesman-Lazer conditions or the monotonicity assumption.
The nonlinearity may be unbounded. The nonlinearity interact, in some sense
with the Steklov spectrum on boundary nonlinearity. The proofs are based on a
priori estimates for possible solutions to a homotopy on suitable trace and
topological degree arguments
Finite reduction and Morse index estimates for mechanical systems
A simple version of exact finite dimensional reduction for the variational
setting of mechanical systems is presented. It is worked out by means of a
thorough global version of the implicit function theorem for monotone
operators. Moreover, the Hessian of the reduced function preserves all the
relevant information of the original one, by Schur's complement, which
spontaneously appears in this context. Finally, the results are
straightforwardly extended to the case of a Dirichlet problem on a bounded
domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential
Equations and Application
Existence of monotonic -solutions for quadratic Volterra functional-integral equations
We study the quadratic integral equation in the space of Orlicz space in the most important case when satisfies the -condition. Considered operators are not compact and then we use the technique of measure of noncompactness associated with the Darbo fixed point theorem to prove the existence of a monotonic, but discontinuous solution. Our present work allows to generalize both previously proved results for quadratic integral equations as well as that for classical equations. Due to different continuity properties of considered operators in Orlicz spaces, we distinguish different cases and we study the problem in the most important case – in such a way to cover all Lebesgue spaces ()
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