Tolokonnikov’s Lemma for Real H ∞ and the Real Disc Algebra

We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space , the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies for all z ∈ . Tolokonnikov’s Lemma for means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in , such that F = [ f f _c ] for some f _c in . In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over , then it has a doubly coprime factorization in . We prove the lemma for the real disc algebra as well. In particular, and are Hermite rings. The work of the first author was supported by Magnus Ehrnrooth Foundation. Received: December 5, 2006. Revised: February 4, 2007.     

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space and a suitable closed subspace of , the Schur complement of A to is defined. The basic properties of are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. To the memory of Professor Mischa Cotlar     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = B_w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w_n) satisfies , the conjugate operator is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an such that is dense in S(H). We generalize the result to more general conjugate maps , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.     

An Operator-valued Berezin Transform and the Class of <Emphasis Type="Italic">n</Emphasis>-Hypercontractions

We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction we associate a positive -valued operator measure dω _{n, T} supported on the closed unit disc in a way that generalizes the above notion of operator-valued Berezin transform. This construction of positive operator measures dω _{n, T} gives a natural functional calculus for the class of n-hypercontractions. We revisit also the operator model theory for the class of n-hypercontractions. The new results here concern certain canonical features of the theory. The operator model theory for the class of n-hypercontractions gives information about the structure of the positive operator measures dω _{n, T} .     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

The C ^*-algebra generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L ²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ^*-algebras associated with points and pairs . Applying a symbol calculus for the abstract unital C ^*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points being the discontinuity points of coefficients. A symbol calculus for the C ^*-algebra is constructed and a Fredholm criterion for the operators is obtained.     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.     

Fong characters and chains of normal subgroups

In this note, we show that if is a π-partial character of the π-separable group is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then has a Fong character α Irr(H) such that for every subgroup , every irreducible constituent of α _H∩N is Fong for N. We also show that if is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of are Fong for M. Received: 21 July 2006 Revised: 17 January 2007     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra as a groupoid crossed product algebra of an arbitrary fixed von Neumann algebra M and the graph groupoid induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid of G has its binary operation, called admissibility. This has concrete local parts , for all e ∈ E(G). We characterize of , induced by the local parts of , for all e ∈ E(G). We then characterize all amalgamated free blocks of . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras , and certain subalgebras (M) of operator-valued matricial algebra . This shows that graph von Neumann algebras identify the key properties of graph groupoids. Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007.     

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

It is proved that the Stokes operator in L^q -space on an infinite cylindrical domain of , , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L ^q where and is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on -boundedness of operator families and on square function estimates involving Muckenhoupt weights.     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

Discrepancy After Adding A Single Set

We show that the hereditary discrepancy of a hypergraph on n points increases by a factor of at most O(log n) when one adds a new edge to .     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006