Animikh Biswas

Extended eigenvalues and the volterra operator

In this paper we consider the integral Volterra operator on the space L 2 (0; 1). We say that a complex number is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV = V X . We show that the set of extended eigenvalues of V is precisely the interval (0;1) and the corresponding eigenvectors may be chosen to be integral operators as well.

Multivariable Generalizations of the Schur Class: Positive Kernel Characterization and Transfer Function Realization

The operator-valued Schur class is defined to be the set of holomorphic functions S mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by S defines a contraction operator between two Hardy Hilbert spaces, S satisfies a von Neumann inequality, a certain operator-valued kernel associated with S is positive-definite, and S can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently, one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence E σ associated with a W*-correspondence E over a W*-algebra \( \mathcal{A} \) together with a *-representation σ of \( \mathcal{A} \). The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szego kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems.

Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

We obtain Gevrey regular mild solutions to the incompressible Navier–Stokes equations in R n with periodic boundary condition in a subset of the variables. The method is based on an extension of Youngs convolution inequality in weighted Lebesgue spaces of measurable functions defined on locally compact abelian groups. This generalizes and provides a unified treatment of the Gevrey regularity result of Foias and Temam in the space periodic case and those of Le Jan and Sznitman and Lemarié–Rieusset in the whole space with no boundary.

A harmonic-type maximal principle in commutant lifting

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.

Dissipation length scale estimates for turbulent flows: A Wiener algebra approach

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier–Stokes system on an

Weighted variants of the Three Chains Completion Theorem

n

Analyticity and Decay Estimates of the Navier-Stokes Equations in Critical Besov Spaces

n-torus. This estimate matches previously known estimates provided that a certain bound on the initial data is satisfied. In particular, it is argued that for two-dimensional (2D) turbulent flows, the initial data is guaranteed to satisfy this hypothesized bound on a significant portion of the 2D global attractor, in which case, the estimate on the radius matches the best known one found in Kukavica (1998). A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms. We note that the method can also be applied with other phase spaces such as that of the functions with square-summable Fourier series, in which case the estimate on the radius matches that of Doering and Titi (1995). It can then similarly be shown that for three-dimensional (3D) turbulent flows, this estimate holds on a significant portion of the 3D weak attractor.

Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted

This work concerns the estimation of a smooth survival function based on doubly censored data. We establish strong consistency and asymptotic normality for a kernel estimator. Moreover, we also obtain an asymptotic expression for the mean integrated squared error, which yields an optimum bandwidth in terms of readily estimable quantities.