A functional calculus in hilbert space based on operator valued analytic functions

A homomorphic map is defined from the algebra of norm bounded analyticN-operator valued functions in the unit disc into the algebra of bounded operators in Hilbert spaces represented as left invariant subspaces ofH ²(N), and the spectral properties of the map are studied. The subclass of functions having norm bound one in the disc is characterized in terms of the power series coefficients. This paper was partially supported by the National Science Foundation under contract NSF GP-5455.     

Generalized Zernike or disc polynomials: An application in quantum optics

An application of the “generalized Zernike or disc polynomials”, recently introduced in the literature, is shown, resorting to the Lie algebra based investigation of the dynamics of quantum systems driven by two-mode interaction Hamiltonians. Further properties of the associated “disc functions” are deduced. Also, a generalization of the disc polynomials towards the Hahn polynomials is suggested.     

Algebraic Functions in the Wiener Algebra

We study algebraic properties of the Wiener algebra of absolutely convergent power series on the closed unit disc. In particular, we prove a form of Weierstrass preparation for algebraic functions in this algebra.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ? R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A ² R+?)^?. It is shown that if the set {λ ∈ ?: dim ker (λ-A)^* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skiï. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If M_λ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then M_λ is cyclic, but disc M_λ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.     

Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In this paper, we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc, giving another proof of the theorem of B. Feigin and the author.     

Uniform Approximation Properties for Spaces of Analytic Functions

We prove that the Banach space of bounded analytic functions and the disc algebra have the uniform bounded approximation property of order p for 1 < p < ∞.     

Analytic and harmonic maps into a topological space

The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.     

Twisted descent algebras and the Solomon-Tits algebra

The notion of descent algebra of a bialgebra is lifted to the Barratt-Joyal setting of twisted bialgebras. The new twisted descent algebras share many properties with their classical counterparts. For example, there are twisted analogs of classical Lie idempotents and of the peak algebra. Moreover, the universal twisted descent algebra is equipped with two products and a coproduct, and there is a fundamental rule linking all three. This algebra is shown to be naturally related to the geometry of the Coxeter complex of type A.     

On The Exact Norm Value Op Some Bmoa Functions

In this paper we construct two large classes of BMOA functions with BMOA norm equal to the H² norm, respecting equal to the H^∞ norm. We also determine all functions in the disc algebra with BMOA norm equal to the Bloch norm.     

C(α) preserving operators on separable Banach spaces

Let T be a bounded linear operator from a separable Banach space X to a Banach space Y. A necessary and sufficient condition on T for the existence of a subspace Z of X such that Z is isomorphic to C(α) and the restriction of T to Z is an isomorphism is given. The special case where X is the disc algebra is then considered and results similar to those previously obtained by the author for C(K) spaces are obtained for the disc algebra. Finally some additional results of the same type are proved for subspaces of C(K) with small annihilator.     

Peak quasisymmetric functions and Eulerian enumeration

Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of cd-index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative cd-index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the g-polynomial for Eulerian posets as an algebra homomorphism.     

On the boundary behaviour of derivatives of functions in the disc algebra

We provide with a simple and explicit construction of a function in the disc algebra, whose derivatives enjoy a disjoint universal property near the boundary. The set of functions with such property is topologically generic, densely lineable, and spaceable.     

The Peak Algebra of the Symmetric Group

The peak set of a permutation is the set {i : (i – 1) < (i) > (i + 1)}. The group algebra of the symmetric group S _n admits a subalgebra in which elements are sums of permutations with a common descent set. In this paper we show the existence of a subalgebra of this descent algebra in which elements are sums of permutations sharing a common peak set. To prove the existence of this peak algebra we use the theory of enriched (P, )-partitions and the algebra of quasisymmetric peak functions studied by Stembridge (Trans. Amer. Math. Soc. 349 (1997) 763–788).     

Higher Order Peak Algebras

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras (Sym(N))_N≥1, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N=1 and N=2). We compute their Hilbert series, introduce and study several combinatorial bases, and establish various algebraic identities related to the multisection of formal power series with noncommutative coefficients. Received November 19, 2004     

Computer extended series solution to viscous flow between rotating discs

The problem of injection (suction) of a viscous incompressible fluid through a rotating porous disc onto a rotating co-axial disc is studied using computer extended series. The universal coefficients in the low Reynolds number perturbation expansion are generated by delegating the routine complex algebra to computer. Various cases leading to specific types of flows are studied. Analytic continuation of the series solution yields results which agree favourably with pure numerical findings up to moderately large Reynolds number. The precise variation of lift as a function of R is established in each case.     

Representations of Hardy Algebras: Absolute Continuity,Intertwiners, and Superharmonic Operators

Suppose T₊(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H ^∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T₊(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H ^∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.     

Colored Posets and Colored Quasisymmetric Functions

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s theory of combinatorial Hopf algebras and its colored analog.