Tensor Products and Functional Calculus of Polynomially Paranormal Operators

We study the connection between the Baum–Connes conjecture for an ample groupoid G G with coefficient A A and the Künneth formula for the K {\mathrm K} -theory of tensor products by the crossed product A ⋊ r G A\rtimes _r G . To do so, we develop the machinery of going-down functors for ample groupoids. As an application, we prove that both the uniform Roe algebra of a coarse space which uniformly embeds in a Hilbert space and the maximal Roe algebra of a space admitting a fibered coarse embedding in a Hilbert space satisfy the Künneth formula. Additionally, we give an example of a space that does not admit a coarse embedding in a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula and provides a stability result for the Künneth formula using controlled K {\mathrm K} -theory. As a byproduct of our methods, we also prove a permanence property for the Baum–Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.

To construct dual frames with good structure for a given frame is a fundamental problem in the theory of frames. The tensor product duals of tensor product frames can provide a rank-one decomposition of bounded antilinear operators between two Hilbert spaces. This paper addresses tensor product dual frames. We derive a necessary and sufficient condition for two tensor product Bessel sequences to be a pair of dual frames; obtain explicit expressions of all dual frames and tensor product dual frames of tensor product frames; and demonstrate the existence of non-tensor product dual frames of tensor product frames.

In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if $$T\in \mathcal{{L(H)}}$$ is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if $$T^{*}$$ has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.

We first determine the kernels of those unitary representations of a locally compact group which are obtained by integrating, inducing and tensoring. We then describe a broad class of group extensions for which the previous results can be used to find (1) the kernel of any representation and (2) necessary and sufficient conditions for these extensions to be maximally almost periodic. Introduction. The primary purpose of this paper is to determine the kernel of any (unitary) representation of a reasonably general locally compact group extension. (See ?4 for a specific description of the exten- sion.) Roughly speaking, the representations of such an extension are obtained by successively tensoring, inducing and integrating (at least in the separable case) representations of certain subgroups. In ?1 we deter- mine the kernel of a direct integral. (This is the only place where we find it necessary to introduce separability assumptions.) This determination has the effect of reducing our problem to the consideration of irreducible representations only. In ?2 we describe the kernel of an induced repre- sentation, thus extending Lemma 2.1 of (11) to the nonseparable case. Regarding the tensor product of representations, it is well known (see (9) and (6, ?17)) that the most useful kernel to determine is that of the tensor product of projective representations or, more accurately, of representa- tions of corresponding central group extensions of the circle. We do this in ?3. Finally, in ?4 we describe the type of extension for which the preceding results solve the problem stated at the beginning. This section is essentially a combination and augmentation of R. J. Blattner's work in (3) and the relevant results of J. M. G. Fell in ?17 of (6). We conclude by indicating how the previous results can be used to find necessary and sufficient conditions for the extension to be maximally almost periodic. Throughout this paper, G will be a locally compact group and all representations will be unitary. If h is a group homomorphism then

A Banach space operator T ∈ B(χ) is left polaroid if for each λ ∈ iso σa(T) there is an integer d(λ) such that asc(T - λ) = d(λ) < ∞ and (T - λ)d(λ)+1χ is closed; T is finitely left polaroid if asc(T - λ) < ∞, (T - λ)χ is closed and dim(T - λ)−1(0) < ∞ at each λ ∈ iso σa(T). The left polaroid property transfers from A and B to their tensor product A ⊗ B, hence also from A and B* to the left-right multiplication operator τAB, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from A and B to their tensor product A⊗B if and only if 0 ∈ iso σa(A⊗B); a similar result holds for τAB for finitely left polaroid A and B*.

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Muller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.

An operator T which maps a Banach space X into itself has the single valued extension property if the only analytic function / which satisfies (l -)/() = 0 is / = 0. Clearly the point spectrum of any operator which does not have the single valued extension property must have nonempty interior. The converse does not hold. However, it is shown below that if o l -T is semi-Fredholm and 0 is an interior point of the point spectrum of , then T does not have the single valued extension property.

Let \(\{T(t)\}_{t\ge 0}\) be a \(C_0\)-semigroup of bounded linear operators on the Banach space X into itself and let A be their infinitesimal generator. In this paper, we show that if T(t) is uniformly ergodic, then A does not have the single valued extension property, which implies that A must have a non-empty interior of the point spectrum. Furthermore, we introduce the local mean ergodic for \(C_0\)-semigroup T(t) at a vector \(x\in X\) and we establish some conditions implying that T(t) is a local mean ergodic at x.

Lower bounds for the complexity of linear functionals in the randomized setting

A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that if and are operators on a complex Hilbert space , then for . It is also shown that if is normal , then . Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product.

The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called “analytically invariant” and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.

Tight frames are extremely useful in applications. A scalable frame was recently introduced as a frame with the property of generating a tight frame by rescaling its frame vectors. In this paper, we consider piecewise scalable frames. We obtain some characterizations for them, and demonstrate that scalability is stable under unitary operators and isomorphisms between two Hilbert spaces. We further obtain a relation between the piecewise scalable frames in Hilbert spaces, and their tensor product

Let $X$ be a Banach space and $T$ be a bounded linear operator on $X$. We denote by $S(T)$ the set of all complex $\lambda \in \mathbb C$ such that $T$ does not have the single-valued extension property at $\lambda $. In this note we prove equality up to $S(T)$ between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl's theorem for operator matrices and multiplier operators.

In this work, we introduce the notion of tensor product of spaces with a generalized 2-inner product (see Definition 5), and we establish several interesting properties (see Proposition5), thereby generalizing the classical properties of the tensor product of inner product spaces. Moreover, we equip this tensor product with a mapping that defines a generalized 2-inner product(see Theorem 3) and, consequently, endow it with a generalized 2-norm (see Theorem 1). In this context, we also define the tensor product of linear operators (see Definition 9) and prove a series of results for example, that the tensor product of two 2-bounded linear operators is again 2-bounded under the tensor product (see Proposition 10).