### Contractions of product density operators of systems of identical fermions and bosons.

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We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

We prove the norm estimates for operator-valued functions on free groups supported on the words with fixed length ($f={\sum}_{\left|w\right|=l}{a}_{w}\otimes \lambda \left(w\right)$). Next, we replace the translations by the free generators with a free family of operators and prove inequalities of the same type.

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 accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible...

We study analytic models of operators of class ${C}_{\xb70}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T\in {C}_{\xb70}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that ${\cong}_{\theta}=A\xb2\u2098\left(\u204e\right)\ominus \theta H\xb2\left(\right)$ and $T\cong {P}_{{}_{\theta}}{M}_{z}{|}_{{}_{\theta}}$, where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their applications...

For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if $x\u2099{\to}^{{\tau}_{\alpha}}x$ implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.

We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators...