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Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and $G$ is the fundamental operator of $(S^*,P^*)$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^*,P^*)$ respectively. Theorem 2 shows that this necessary condition is sufficient too provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for $Gamma$-contractions are then applied to tetrablock contractions to figure out when two pairs $(F_1, F_2)$ and $(G_1, G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A, B, P)$ and its adjoint $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4.
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Admissible vectors lead to frames or coherent states under the action of a group by means of square integrable representations. This work shows that admissible vectors can be seen as weights with central support on the (left) group von Neumann algebra. The analysis involves spatial and cocycle derivatives, noncommutative $L^p$-Fourier transforms and Radon-Nikodym theorems. Square integrability confine the weights in the predual of the algebra and everything may be written in terms of a (right selfdual) bounded element.
The epsilon-enlargement of a maximal monotone operator is a construct similar to the Br{o}ndsted and Rocakfellar epsilon-subdifferential enlargement of the subdifferential. Like the epsilon-subdifferential, the epsilon-enlargement of a maximal monotone operator has practical and theoretical applications. In a recent paper in Journal of Convex Analysis Burachik and Iusem studied conditions under which a maximal monotone operator is non-enlargeable, that is, its epsilon-enlargement coincides with the operator. Burachik and Iusem studied these non-enlargeable operators in reflexive Banach spaces, assuming the interior of the domain of the operator to be nonempty. In the present work, we remove the assumption on the domain of non-enlargeable operators and also present partial results for the non-reflexive case.
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.
We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger-Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $mathcal A_1$, in the chain $mathcal A_0 subseteq mathcal A_1 subseteq ldots subseteq mathcal A_{infty}$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $mathcal A_k$ is finite or not. In particular, a Berger-Shaw-type theorem fails to be true for members of~$mathcal A_{infty}$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.
In this article we introduce a new class of Rolewicz-type operators in l_p, $1 le p < infty$. We exhibit a collection F of cardinality continuum of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.
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