We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.

We describe the C*-algebra associated with the finite sections discretization of truncated Toeplitz operators on the model space K2u where u is an infinite Blaschke product. As consequences, we get a stability criterion for the finite sections discretization and results on spectral and pseudospectral approximation.

We consider the norm closure 𝔄 of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact manifold X with boundary ∂X. Assuming that all connected components of X have nonempty boundary, we show that K₁(𝔄) ≃ K₁(C(X)) ⊕ ker χ, where χ: K₀(C₀(T*Ẋ)) → ℤ is the topological index, T*Ẋ denoting the cotangent bundle of the interior. Also K₀(𝔄) is topologically determined. In case ∂X has torsion free K-theory, we get K₀(𝔄) ≃ K₀(C(X)) ⊕ K₁(C₀(T*Ẋ)).

To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm...

If G is a discrete group, the algebra CD(G) of convolution dominated operators on l²(G) (see Definition 1 below) is canonically isomorphic to a twisted L¹-algebra $l¹(G,l^{\infty }\left(G\right),T)$. For amenable and rigidly symmetric G we use this to show that any element of this algebra is invertible in the algebra itself if and only if it is invertible as a bounded operator on l²(G), i.e. CD(G) is spectral in the algebra of all bounded operators. For G commutative, this result is known (see [1], [6]), for G noncommutative discrete...

A review of recent reflexivity and hyperreflexivity results is presented. We concentrate particularly on a finite-dimensional situation, Toeplitz operators and partial isometries. Open problems in this area are given.

The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane are investigated. The dichotomic behavior (transitive or reflexive) of these subspaces is shown. It refers to the similar dichotomic behavior for subspaces of Toeplitz operators on the Hardy space on the unit disc. The isomorphism between the Hardy spaces on the unit disc and the upper half-plane is used. To keep weak* homeomorphism between $L^{\infty }$ spaces on the unit circle and the real line...

It was shown that the space of Toeplitz operators perturbated by finite rank operators is 2-hyperreflexive.

Projections onto the spaces of all Toeplitz operators on the N-torus and the unit sphere are constructed. The constructions are also extended to generalized Toeplitz operators and applied to show hyperreflexivity results.

This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product...

Subspaces of Toeplitz operators on the Hardy spaces over a multiply connected region in the complex plane are investigated. A universal covering map of such a region and the group of automorphisms invariant with respect to the covering map connect the Hardy space on this multiply connected region with a certain subspace of the classical Hardy space on the disc. We also present some connections of Toeplitz operators on both spaces from the reflexivity point of view.

Based on the results in A. Feintuch (1989), this work sheds light upon some interesting properties of strongly asymptotically Toeplitz and Hankel operators, and relations between these two classes of operators. Indeed, among other things, two main results here are (a) vanishing Toeplitz and Hankel operators forms an ideal, and (b) finding the distance of a strongly asymptotically Toeplitz operator from the set of vanishing Toeplitz operators.

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products...

We study general continuity properties for an increasing family of Banach spaces $S_w^p$ of classes for pseudo-differential symbols, where $S_w^{\infty }=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\mathrm{Op}\left(S_w^p\right)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ and $S_w^p·S_w^q\subset S_w^r$, provided $1/p+1/q=1/r$. If instead $1/p+1/q=1+1/r$, then $S_w^{p}w*S_w^q\subset S_w^r$. By modifying the definition of the $S_w^p$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.