The epsilon-pseudospectrum ?(a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a. Suppose for some_x000f_ epsilon > 0 and a,b \in A, ?(ax) = ?(bx) for all x \in A. It is shown that a = b if: (I) a is invertible. (ii) a is Hermitian idempotent. (iii) a is the product of a Hermitian idempotent and an invertible element. (iv) A is semisimple and a is the product of an idempotent and an invertible element. (v) A = B (X) for a Banach space X. (vi) A is a C*-algebra. (vii) A is a commutative semisimple Banach algebra.

A famous theorem of Norbert Wiener states that for a function f in L^1(R), span of translates f(x?a) with complex coefficients is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. That is the ideal generated by f in L^1(R) is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. This theorem is well known as the Wiener Tauberian theorem. This theorem has been extended to abelian groups. The hypothesis (in the abelian case) is on a Haar integrable function which has nonvanishing Fourier transform on all unitary characters. However, back in 1955, Ehrenpreis and Mautner observed that Wiener Tauberian theorem fails even for the commutative Banach algebra of integrable radial functions on SL(2,R). In this talk we shall discuss about a genuine analogue of the theorem for real rank one, connected noncompact semisimple Lie groups with finite centre.

http://www.math.iitb.ac.in/~seminar/Starting with a positive definite kernel $K$ defined on a bounded open connected subset $\Omega$ of $\mathbb C^d,$ we give several canonical constructions for producing new positive definite kernels on $\Omega,$ possibly taking values in $Hom(E)$ for some normed linear space $E$ of dimension $d.$ Specifically, this includes the curvature defined as the $d\times d$ matrix of real analytic functions $$\big ( \!\! \big ( \tfrac{\partial}{\partial_i \bar{\partial}_j} \log K \big ) \!\!\big ).$$ These kernels define an inner product on a submodule (over the polynomial ring) functions holomorphic on $\Omega.$ The completion is a Hilbert space on which the polynomials act by point-wise multiplication making it into a "Hilbert module". We will discuss hereditary properties, sub and quotient of these Hilbert modules.

In recent years multiple branch of mathematics have been 'quantised', in other words - made noncommutative. In this talk we will show how this noncommutation procedure is applied to the field of probability, more precisely - to stochastic calculus. We will mainly focus on the ideas leading us to quantising the notion of Wiener chaos via multiple Wiener integrals. This is joint work with Prof. J. Martin Lindsay.