Recall some notations and terminology from [4]. For closed subspaces M, L of a Banach space X we write M e ⊂L (M is essentially contained in L ) if there exists a finite-dimensional subspace F ⊂ X such that M ⊂ L + F . Equivalently, dim M/(M ∩L) = dim(M + L)/L < ∞. Similarly we write M e =L if M e ⊂L and L e ⊂M . For a (bounded linear) operator T ∈ L(X) write R∞(T ) = ∞n=0 R(T) and N∞(T ) = ⋃∞ ...

We present the OSp(2, 2|6) symmetric algebraic curve for the AdS 3 /CF T 4 duality recently proposed in arXiv:0806.1218. It encodes all classical string solutions at strong t'Hooft coupling and the full two loop spectrum of long single trace gauge invariant operators in the weak coupling regime. This construction can also be used to compute the complete superstring semi-classical spectrum aroun...

The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra Uq[ ̂ sl(2|1)]. We give the bosonization of the boundary states. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy. E-mail: [email protected] E-mail:...

for some positive constants m,M and all x, t and ξ. The basic example is the heat, or diffusion, equation ut − ∆u = 0 , which we start by solving, first for x in an interval and then in R. We then show that in both situations the solutions fit into an abstract framework of what is called a semi-group of contraction operators. We then discuss some properties of solutions of general parabolic equ...

a subset of real Euclidean n-space, En, by (p, q) → F (p − q) = p ◦ q. In this note we shall be concerned with a representation T (.) of π as a semi-group of bounded linear operators on a Banach space X. More particularly, we suppose that postulates P1, P2, P3, P5, and P6 of chapter 25 of (2) are satisfied so that, by Theorem 25.3.1 of that book, there is a continuous function, f(.), defined on...

for suitable banach spaces $x$ and $y$ with schauder decompositions and‎ ‎a suitable closed subspace $mathcal{m}$ of some compact operator space from $x$ to $y$‎, ‎it is shown that the strong banach-saks-ness of all evaluation‎ ‎operators on ${mathcal m}$ is a sufficient condition for the weak‎ ‎banach-saks property of ${mathcal m}$, where for each $xin x$ and $y^*in‎ ‎y^*$‎, ‎the evaluation op...