An operator T acting on a Banach space X satisfies the property (UWE) if σa(T ) r σSF− + (T ) = E(T ), where σa(T ) is the aproximate point spectrum of T , σ SF− + (T ) is the upper semi-Weyl spectrum of T and E(T ) is the set of all eigenvalues of T that are isolated in the spectrum σ(T ) of T . In this paper, we introduce and study two new spectral properties, namely (VE) and (VEa ), in conne...

Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensi...

Mark Kac introduced a method for calculating the distribution of the integral Av= ∫ T 0 v(Xt) dt for a function v of a Markov process (Xt; t¿0) and a suitable random time T , which yields the Feynman–Kac formula for the moment-generating function of Av. We review Kac’s method, with emphasis on an aspect often overlooked. This is Kac’s formula for moments of Av, which may be stated as follows. F...

In this paper, we state some results on product of operators with closed ranges and we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by using some block operator matrix techniques, we nd explicit solution of the operator equation $TXS^*-SX^*T^*= A$.

The purpose of this paper is to introduce the concept of soft double fuzzy semi-topogenous order. Firstly, we give the definition of soft double fuzzy semi-topogenous order. Secondly, we induce a soft double fuzzy topology from a given soft double fuzzy semi-topogenous order by using soft double fuzzy interior operator.

This paper presents a variation of constants formula for the system of functional parabolic partial differential equations ∂u(t, x) ∂t = D∆u+ Lut + f(t, x), t > 0, u ∈ R ∂u(t, x) ∂η = 0, t > 0, x ∈ ∂Ω u(0, x) = φ(x) u(s, x) = φ(s, x), s ∈ [−τ, 0), x ∈ Ω . Here Ω is a bounded domain in Rn, the n × n matrix D is block diagonal with semi-simple eigenvalues having non negative real part, the operat...

In this paper, we show that an unbounded weakly S0-demicompact linear operator T, introduced in [18], acting on a Banach space, can be characterized by some measures of weak noncompactness. Moreover, our results are illustrated to discuss the relationship with Fredholm and upper semi-Fredholm operators as well stability essential spectrum T.

Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...