On commuting operator exponentials, II

We prove that, under sufficient conditions on the spectra, $$ e^Me^N\subseteq e^Ne^M\Rightarrow MN\subseteq NM , $$ where N is an unbounded normal operator and M is a bounded normal operator in the Hilbert space.     

Splitting for subalgebras of tensor products

We prove splitting results for subalgebras of tensor products of operator algebras. In particular, any -algebra s.t. is a tensor product provided is simple and nuclear.

     

Equivalence of quotient Hilbert modules-II

For any open, connected and bounded set , let be a natural function algebra consisting of functions holomorphic on . Let be a Hilbert module over the algebra and let be the submodule of functions vanishing to order on a hypersurface . Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module in the case of . In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.

     

On the similarity problem for polynomially bounded operators on hilbert space

Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S ^?1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining $$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$ onl ² ⊕l ², whereS is the shift and Γ _f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR _f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate $$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$ whereSTS ^?1 is a contraction and assuming \(\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty \) wheneverp is an analytic polynomial on the disc.     

Invariance theory for infinite dimensional linear control systems

In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system , whereA is the infinitesimal generator of aC ₀-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space intoX. WhenS D(A^*) is dense inS , it is shown that a necessary (but insufficient) condition for holdability is (1): . A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, B andS+ B are closed.In the unbounded case, our results are illustrated by the shift semigroup and by the heat equation on an infinite rod with distributed controls. In the bounded case, our example is an integro-differential control system.Research sponsored by the National Research Council of Canada under Grant A7271.Research sponsored by the National Research Council of Canada under Grant A4641.     

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Mycielski ideal and the perfect set theorem

We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If is an analytic set all projections of which are uncountable, then there is a perfect set a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets.

     

Spectrum of the Kerzman–Stein Operator for Model Domains

For a domain , the Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator acting on L²(b), which is defined with respect to Euclidean measure. In this paper we compute the spectrum of the Kerzman-Stein operator for three domains whose boundaries consist of two circular arcs: a strip, a wedge, and an annulus. We also treat the case of a domain bounded by two logarithmic spirals.     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Spectrally bounded -derivations on Banach algebras

Applying the density theorem on algebras with -derivations, we show that if a -derivation of a unital Banach algebra is spectrally bounded, then . Also, if and only if , where denotes the spectral radius of .

     

Intermediate subfactors with no extra structure

If are type II factors with and we show that restrictions on the standard invariants of the elementary inclusions , , and imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto and do not commute, then is or . In the former case is the fixed point algebra for an outer action of on and the angle is , and in the latter case the angle is and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

     

On finiteness of chains of intermediate rings

An extension of integral domains is said to have the “finite length of intermediate chains of domains” property (for short FICP) if each chain of intermediate rings between R and S is finite. The main purpose of this paper is to characterize when has FICP in case R ^* (the integral closure of R in S) is a finite dimensional semilocal domain. This generalizes a theorem due to Gilmer, in which S is the quotient field of R. Examples illustrating the sharpness and the limits of our results are settled.      

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.

     

The $$\overline \partial $$-Neumann operator on Lipschitz q-pseudoconvex domains

On a bounded q-pseudoconvex domain Ω in ? ⁿ with a Lipschitz boundary, we prove that the \(\overline \partial \)-Neumann operator N satisfies a subelliptic (1/2)-estimate on Ω and N can be extended as a bounded operator from Sobolev (?1/2)-spaces to Sobolev (1/2)-spaces.     

Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications

The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to

$$\Delta_{\mathbb{H}^n}^2 u=\frac{f(\xi,u)}{\rho(\xi)^a}\,\,\text{ in}\Omega,\,\, u|_{\partial\Omega}=0=\left.\frac{\partial u}{\partial\nu}\right|_{\partial\Omega},$$

where \({0\in \Omega \subseteq \mathbb{H}^4}\) is a bounded domain, \(0 \leq a \leq Q,\,(Q=10).\) The special feature of this problem is that it contains an exponential nonlinearity and singular potential.

     

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$ . Then, $ AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.$      

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.