A famous result of S. Kwapień asserts that a linear operator from Banach space to Hilbert is absolutely 1-summing whenever its adjoint q-summing for some 1 ≤ q < ∞; this was recently extended Lipschitz operators by Chen and Zheng. In the present paper we show Kwapień’s Chen-Zheng’s theorems hold in very relaxed nonlinear environment, under weaker hypotheses. Even when restricted original case, ...

In this paper we improve a result of W. B. Johnson and G. Schechtman by proving that the p-summing norm of any operator with n-dimensional domain can be well-approximated using C(p)n logn(log logn)2 vectors if 1 < p < 2, and using C(p)np/2 logn if 2 < p <∞. 1. p-summing norms Throughout this paper we will follow notations of N. Tomczak-Jaegermann [To]. Definition 1.1. Let X and Y be Banach spac...

We give a simple proof of Bourgain's disc algebra version of Grothendieck's theorem, i.e. that every operator on the disc algebra with values in L 1 or L 2 is 2-absolutely summing and hence extends to an operator defined on the whole of C. This implies Bourgain's result that L 1 /H 1 is of cotype 2. We also prove more generally that L r /H r is of cotype 2 for 0 < r < 1.

Given p ≥ 1, we denote by Cp the class of all Banach spaces X satisfying the equality Kp(Y,X) = Πp(Y,X) for every Banach space Y , Kp (respectively, Πp) being the operator ideal of p-compact operators (respectively, of operators with p-summing adjoint). If X belongs to Cp, a bounded set A ⊂ X is relatively p-compact if and only if the evaluation map U∗ A : X ∗ −→ ∞(A) is p-summing. We obtain p-...

We consider tractable representations of probability distributions and the polytime operations they support. In particular, we consider a recently proposed arithmetic circuit representation, the Probabilistic Sentential Decision Diagram (PSDD). We show that PSDDs support a polytime multiplication operator, while they do not support a polytime operator for summing-out variables. A polytime multi...

Let \(X\), \(Y\) and \(Z\) be Banach spaces let \(U\) a subspace of \(\mathcal{L}(X^*,Y)\), the space all operators from \(X^*\) to \(Y\). An operator \(S\colon U \to Z\) is said \((\ell^s_p,\ell_p)\)-summing (where \(1\leq p <\infty\)) if there constant \(K\geq 0\) such that \(\left( \sum_{i=1}^n \|S(T_i)\|_Z^p \right)^{1/p}\le K\sup_{x^* \in B_{X^*}} \left(\sum_{i=1}^n \|T_i(x^*)\|_Y^p\right...

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a ”good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessib...

We discuss the ϕ 6 theory defined in D = 2 + 1-dimensional space-time and assume that the system is in equilibrium with a thermal bath at temperature β −1. We use the 1/N expansion and the method of the composite operator (CJT) for summing a large set of Feynman graphs.We demonstrate explicitly the Coleman-Mermin-Wagner theorem at finite temperature.

We discuss the ϕ 6 theory defined in D = 2 + 1-dimensional space-time and assume that the system is in equilibrium with a thermal bath at temperature β −1. We use the 1/N expansion and the method of the composite operator (CJT) for summing a large set of Feynman graphs.We demonstrate explicitly the Coleman-Mermin-Wagner theorem at finite temperature.

We consider summation of consecutive values φ(v), φ(v+1), . . . , φ(w) of a meromorphic function φ(z) where v, w ∈ ZZ. We assume that φ(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found – if it exists – by the Accurate Summation algorithm, or alternatively, by Gosper’s algorithm when ordL = 1)...