Quasi-Banach operator ideals with a very strange trace

As shown by S. Lord, F. Sukochev, and D. Zanin (see [7]), the theory of singular traces is well understood for operators on the Hilbert space. The situation turns out to be completely different in the Banach space setting. Indeed, quite strange phenomena may occur. We will construct quasi-Banach operator ideals ${\mathfrak A}$ A with seemingly contradictory properties: On the one hand, ${\mathfrak A}$ A supports a continuous trace τ that vanishes at all finite rank operators, which means that τ is singular. On the other hand, ${\mathfrak A}$ A contains the identity map I _Z of an infinite-dimensional Banach space Z and τ (I _Z ) =  1. This implies that there exist operators ${T \in \mathfrak A (Z)}$ T ∈ A ( Z ) such that ${\tau (T^n) = 1}$ τ ( T n ) = 1 for ${n = 1,2,{\dots} \;}$ n = 1 , 2 , ? , which is impossible for singular traces in the case of a Hilbert space. As most counterexamples, the new operator ideals have no useful application. They provide, however, a deeper insight into the philosophy of traces.     

Operator Ideals on Non-commutative Function Spaces

Suppose X and Y are Banach spaces, and \({{\mathcal{I}}}\) , \({{\mathcal{J}}}\) are operator ideals. compact operators). Under what conditions does the inclusion \({\mathcal{I}(X,Y) \subset \mathcal{J}(X,Y)}\) , or the equality \({\mathcal{I}(X,Y)\,=\,\mathcal{J}(X,Y)}\) , hold? We examine this question when \({\mathcal{I}, \mathcal{J}}\) are the ideals of Dunford–Pettis, strictly (co)singular, finitely strictly singular, inessential, or (weakly) compact operators, while X and Y are non-commutative function spaces. Since such spaces are ordered, we also address the same questions for positive parts of such ideals.     

Extension and Lifting of Operators and Polynomials

We study the problem of extension and lifting of operators belonging to certain operator ideals, as well as that of their associated polynomials and holomorphic functions. Our results provide a characterization of ${\mathcal{L}_1}$ and ${\mathcal{L}_\infty}$ -spaces that includes and extends those of Lindenstrauss-Rosenthal [32] using compact operators and González-Gutiérrez [23] using compact polynomials. We display several examples to show the difference between extending and lifting compact (resp. weakly compact, unconditionally convergent, separable and Rosenthal) operators to operators of the same type. Finally, we show the previous results in a homological perspective, which helps the interested reader to understand the motivations and nature of the results presented.     

WQ*-algebras of measurable operators

Every C*-algebra $\mathfrak{A}$ has a faithful *-representation π in a Hilbert space $\mathcal{H}$ . Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.     

Parabolic Non-Automorphism Induced Toeplitz-Composition C*-Algebras with Piece-wise Quasi-Continuous symbols

In this paper we consider the C*-algebra $C^{*}(\{C_{\varphi }\}\cup \mathcal T (PQC(\mathbb T )))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$ . This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.     

A Hilbert Module Approach to the Haagerup Property

We develop a Hilbert module version of the Haagerup property for general C*-algebras ${{\mathcal{A} \subseteq \mathcal{B}}}$ . We show that if ${\alpha : \Gamma \curvearrowright \mathcal{A}}$ is an action of a countable discrete group Γ on a unital C*-algebra ${\mathcal{A}}$ , then the reduced C*-algebra crossed product ${\Gamma \ltimes _{\alpha, r} \mathcal{A}}$ has the Hilbert ${\mathcal{A}}$ -module Haagerup property if and only if the action α has the Haagerup property. We are particularly interested in the case when ${\mathcal{A} = C(X)}$ is a unital commutative C*-algebra. We compare the Haagerup property of such an action ${\alpha: \Gamma \curvearrowright C(X)}$ with the two special cases when (1) Γ has the Haagerup property and (2) Γ is coarsely embeddable into a Hilbert space. We also prove a contractive Schur mutiplier characterization for groups coarsely embeddable into a Hilbert space, and a uniformly bounded Schur multiplier characterization for exact groups.     

Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.     

The Eigenfunctions of the Hilbert Matrix

For each noninteger complex number ??, the Hilbert matrix $$H_\lambda= \biggl( \frac{1}{n+m+\lambda} \biggr)_{n,m\geq0}$$ defines a bounded linear operator on the Hardy spaces $\mathcal{H}^{p}$ , 1<p<??, and on the Korenblum spaces $\mathcal{A}^{-\tau}$ , ??>0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex ?? results by Hill and Rosenblum for real ??. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.     

Supercyclicity in Spaces of Operators

A sufficient criterion for the map \({C_{A, B}(S) = ASB}\) to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map \({C_{T}(V) = TVT^*}\) is shown to be norm-supercyclic on the algebra \({\mathcal{K}(H)}\) of all compact operators, COT-supercyclic on the real subspace \({\mathcal{S}(H)}\) of all self-adjoint operators and weak*-supercyclic on \({\mathcal{L}(H)}\) of all bounded operators on H. Examples including operators of the form \({C_{B_w, F_\mu}}\) are provided, where B_w and \({F_\mu}\) are respectively backward and forward shifts on Banach sequence spaces.     

Embeddings of Müntz Spaces: Composition Operators

Given a strictly increasing sequence ${\Lambda = (\lambda_n)}$ of nonnegative real numbers, with ${\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty}$ , the Müntz spaces ${M_\Lambda^p}$ are defined as the closure in L ^p ([0, 1]) of the monomials ${x^{\lambda_n}}$ . We discuss how properties of the embedding ${M_\Lambda^2\subset L^2(\mu)}$ , where?μ is a finite positive Borel measure on the interval [0, 1], have immediate consequences for composition operators on ${M^2_\Lambda}$ . We give criteria for composition operators to be bounded, compact, or to belong to the Schatten–von Neumann ideals.     

Unbounded Induced Representations of ∗-Algebras

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra ${{\mathcal{A}}}$ onto a unital *-subalgebra ${{\mathcal{B}}}$ are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras ${{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g$ for which the *-subalgebra ${{\mathcal{B}}}:={{\mathcal{A}}}_e$ is commutative. Then the canonical projection $p:{{\mathcal{A}}}\to{{\mathcal{B}}}$ is a conditional expectation and there is a partial action of the group G on the set ${{\mathcal{B}}}p$ of all characters of ${{\mathcal{B}}}$ which are nonnegative on the cone $\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.$ The complete Mackey theory is developed for *-representations of ${{\mathcal{A}}}$ which are induced from characters of ${{\widehat{{{\mathcal{B}}}}^+}}.$ Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras ${{\mathcal{A}}}$ is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.     

Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space $\mathcal{H }$ . Without the condition that either $\mathcal{H }$ is finite dimensional or the sum of $n$ operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.     

Poset embeddings of Hilbert functions

For a standard graded algebra $R$ , we consider embeddings of the poset of Hilbert functions of $R$ -ideals into the poset of $R$ -ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements–Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.     

Approximate Unitary Equivalence to Skew Symmetric Operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called skew symmetric if \(T\) can be represented as a skew symmetric matrix relative to some orthonormal basis for \(\mathcal {H}\) . In this paper, we study the approximation of skew symmetric operators and provide a \(C^*\) -algebra approach to skew symmetric operators. We classify up to approximate unitary equivalence those skew symmetric operators \(T\in \mathcal {B(H)}\) satisfying \(C^*(T)\cap \mathcal {K(H)}=\{0\}\) . This is used to characterize when a unilateral weighted shift with nonzero weights is approximately unitarily equivalent to a skew symmetric operator.     

On *-Representations of Polynomial Algebras in Quantum Matrix Spaces of Rank 2

In this paper we study *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices Mat₂ and on quantum complex symmetric matrices \(\mathrm{Mat_2^{sym}}\) . For the second algebra we classify all irreducible *-representations by bounded operators in a Hilbert space (up to a unitary equivalence). Moreover, we present a construction of *-representations of the above algebras which enables to obtain the full list of *-representations (sometimes by passing to subrepresentations).     

Generic non-selfadjoint Zakharov–Shabat operators

In this paper we develop tools to study within a family of non-selfadjoint operators $L(\varphi )$ depending on a parameter $\varphi $ in a real Hilbert space, those with (partially) simple spectrum. As a case study we consider the Zakharov–Shabat operators $L(\varphi )$ appearing in the Lax pair of the focusing NLS on the circle. In particular, the main result implies that the set of potentials $\varphi $ of Sobolev class $H^N$ , $N\ge 0$ , so that all non real eigenvalues of $L(\varphi )$ are simple, is path connected and dense.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Triplets of Closely Embedded Dirichlet Type Spaces on the Unit Polydisc

We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted $L^2$ spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces $\mathcal{D }_\alpha (\mathbb{D }^N)$ of holomorphic functions on the unit polydisc $\mathbb{D }^N$ and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space $H^2(\mathbb{D }^N)$ through a scale of Dirichlet type spaces and Bergman type spaces.