Riesz Projection Research Articles

Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces

Let X be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Taf=P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X=L2 and show that, under certain assumptions on the space X, the Toeplitz operator Ta is bounded (resp., compact) if and only if a∈L∞ (resp., a=0). Moreover, aL∞≤TaB(H[X])≤PB(X)aL∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.

The multiplicative Hilbert matrix

It is observed that the infinite matrix with entries (mnlog⁡(mn))−1 for m,n≥2 appears as the matrix of the integral operator Hf(s):=∫1/2+∞f(w)(ζ(w+s)−1)dw with respect to the basis (n−s)n≥2; here ζ(s) is the Riemann zeta function and H is defined on the Hilbert space H02 of Dirichlet series vanishing at +∞ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus T∞. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals π and that it has a purely continuous spectrum which is the interval [0,π]; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix (m1/pn(p−1)/plog⁡(mn))−1 has norm π/sin⁡(π/p) when acting on ℓp for 1<p<∞. However, the multiplicative Hilbert matrix fails to define a bounded operator on H0p for p≠2, where H0p are Hp spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol ∑n≥2(log⁡n)−1n−s−1/2 of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus T∞.

RIESZ PROJECTIONS FOR A NON-HYPONORMAL OPERATOR

J. G. Stampfli proved that if a bounded linear operator $T$ on a Hilbert space $\mathscr H$ satisfies ($G_1$) property, then the Riesz projection $P_{\lambda}$ associated with $\lambda\in{\rm iso}\sigma(T)$ is self-adjoint and $P_{\lambda}\mathscr{H}=(T - \lambda)^{-1}(0)=(T^{*} - \bar{\lambda})^{-1}(0)$. In this note we show that Stampfli's result is generalized to an nilpotent extension of an operator having ($G_1$) property.

A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions

A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent (λ−T)−1ϕ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X′-valued holomorphic function for any ϕ∈X, even when T has a continuous spectrum on R, where X′ is a dual space of X. The rigged Hilbert space consists of three spaces X⊂H⊂X′. A generalized eigenvalue and a generalized eigenfunction in X′ are defined by using the analytic continuation of the resolvent as an operator from X into X′. Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.

Geometry of the spectral semidistance in Banach algebras

Let A be a unital Banach algebra over ℂ, and suppose that the nonzero spectral values of a and b ∈ A are discrete sets which cluster at 0 ∈ ℂ, if anywhere. We develop a plane geometric formula for the spectral semidistance of a and b which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that a and b are quasinilpotent equivalent if and only if all the Riesz projections, p(α, a) and p(α, b), correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space X in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu’s geometric formula (which requires the knowledge of the local spectra of the operators at each 0 ≠ x ∈ X). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results.

Sharp estimates for holomorphic functions on the unit ball of ℂn

The purpose of the paper is to establish sharp estimates for the Hilbert transform and the Riesz projection on the unit sphere of . The proof rests on the existence of certain superharmonic functions on satisfying appropriate majorization conditions.

Система Дирака с недифференцируемым потенциалом и антипериодическими краевыми условиями

Saratov State University, Russia, 410012, Saratov, Astrahanskaya st., 83, KornevVV@info.sgu.ru, KhromovAP@info.sgu.ruTheobjectofthepaperisDiracsystemwithantiperiodicboundaryconditionsandcomplex-valuedconditionspotential.Anewmethodis suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transformoperators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it isproved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensionalvector-functions since eigenvalues may be multiple. The structure of Riesz projection operators is also studied. The results of thepaper can be used in spectral problems for equations with partial derivatives of the 1-st order containing involution.Key words: Dirac system, spectrum, asymptotics, Riesz basis.

Finite spectra and quasinilpotent equivalence in Banach algebras

This paper further investigates the implications of quasinilpotent equivalence for (pairs of) elements belonging to the socle of a semisimple Banach algebra. Specifically, not only does quasinilpotent equivalence of two socle elements imply spectral equality, but also the trace, determinant and spectral multiplicities of the elements must agree. It is hence shown that quasinilpotent equivalence is established by a weaker formula (than that of the spectral semidistance). More generally, in the second part, we show that two elements possessing finite spectra are quasinilpotent equivalent if and only if they share the same set of Riesz projections. This is then used to obtain further characterizations in a number of general, as well as more specific situations. Thirdly, we show that the ideas in the preceding sections turn out to be useful in the case of C*-algebras, but now for elements with infinite spectra; we give two results which may indicate a direction for further research.

A lower bound in Nehari’s theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari’s theorem (i.e., if H ψ is a bounded Hankel form on H 2(D d) with analytic symbol ψ, then there is a function φ in L ∞(T d ) such that ψ is the Riesz projection of g4) is known to hold on the polydisc D d for d > 1. A method proposed in Helson’s last paper is used to show that the constant C d in the estimate ‖φ‖∞ ≤ C d ‖H ψ ‖ grows at least exponentially with d; it follows that there is no analogue of Nehari’s theorem on the infinite-dimensional polydisc.

A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that V∈L2([0,π];dx), we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators −d2/dx2+V in L2([0,π];dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators −d2/dx2+V in Lp([0,π];dx), p∈(1,∞).

Logarithmic estimates for the Hilbert transform and the Riesz projection

We study sharp Llog L inequalities for the Hilbert transform and the Riesz projection acting on vector-valued functions defined on the unit circle.

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Abstract We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue– Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten–von- Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

Formula omitted] to [formula omitted] constants for Riesz projections

The norm of the Riesz projection from L ∞ ( T n ) to L p ( T n ) is considered. It is shown that for n = 1 , the norm equals 1 if and only if p ⩽ 4 and that the norm behaves asymptotically as p / ( π e ) when p → ∞ . The critical exponent p n is the supremum of those p for which the norm equals 1. It is proved that 2 + 2 / ( 2 n − 1 ) ⩽ p n < 4 for n > 1 ; it is unknown whether the critical exponent for n = ∞ exceeds 2.

Bari–Markus property for Riesz projections of 1D periodic Dirac operators

AbstractThe Dirac operators equation image with L2‐potentials equation image considered on [0, π] with periodic, antiperiodic or Dirichlet boundary conditions (bc), have discrete spectra, and the Riesz projections equation image are well‐defined for |n | ≥ N if N is sufficiently large. It is proved that equation image where P0 n, n ∈ ℤ, are the Riesz projections of the free operator.Then, by the Bari–Markus criterion, the spectral Riesz decompositions equation image converge unconditionally in L2 (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type

The Hilbert matrix induces a bounded operator on most Hardy and Bergman spaces, as was shown by Diamantopoulos and Siskakis. We generalize this for any Hankel operator on Hardy spaces by using a result of Hollenbeck and Verbitsky on the Riesz projection and also compute the exact value of the norm of the Hilbert matrix. Using a new technique, we determine the norm of the Hilbert matrix on a wide range of Bergman spaces.

Continuity versus boundedness of the spectral factorization mapping

This paper characterizes the Banach algebras of continuous functions on which the spectral factorization mapping S is continuous or bounded. It is shown that S is continuous if and only if the Riesz projection is bounded on the algebra, and that S is bounded only if the algebra is isomorphic to the algebra of continuous functions. Conse- quently, S can never be both continuous and bounded, on any algebra under consideration. 1. Introduction. The operation by which a functionf on the unit circle is written as f( ) = f+( )f+( ) for all 2 T :=fz2 C :jzj = 1g, with an outer functionf+, is known as spectral factorization. This operation arises in many dierent

Generalized Riesz projections and Toeplitz operators

Generalized Riesz projections and Toeplitz operators

The matrix-valued [formula omitted] corona problem in the disk and polydisk

In this paper, we consider the matrix-valued H p corona problem in the disk and polydisk. The result for the disk is rather well-known, and is usually obtained from the classical Carleson Corona Theorem by linear algebra. Our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solution. In particular, we were able to improve the estimate found in the recent work of Trent in [J. Funct. Anal. 189 (2002) 267–282]. Note that, the solution of the H ∞ matrix corona problem in the disk can be easily obtained from the H 2 corona problem either by factorization, or by the Commutant Lifting Theorem. The H p corona problem in the polydisk was originally solved by Lin in [Bull. Sci. Math. 110(2) (1986) 69–84, Trans. Amer. Math. Soc. 341 (1994) 371–375]. The solution used Koszul complexes and was rather complicated because one had to consider higher order ∂ ¯ -equations. Our proof is more transparent and it improves upon Lin's result in several ways. First, we deal with the more general matrix corona problem. Second, we were able to show that the norm of the solution is independent of the number of generators. Additionally, we illustrate that the norm of the solution of the H 2 corona problem in the polydisk D n grows at most proportionally to n . Our approach is based on one that was originated by Andersson in [Math. Z. 201 (1989) 121–130]. In the disk it essentially depends on Green's Theorem and duality to obtain the estimate. In the polydisk we use Riesz projections to reduce the problem to the disk case.

Riesz projections for a class of Hilbert space operators

A Hilbert space operator T, T ∈ B ( H ) , is totally hereditarily normaloid, T ∈ THN , if every part and (also) every invertible part of T is normaloid. The class THN is large. It is proved that if a T ∈ THN is such that the isolated eigenvalues of T are normal, then the Riesz projection P λ associated with a λ ∈ iso σ( T) is self-adjoint and P λ H = ( T - λ ) - 1 ( 0 ) = ( T ∗ - λ ¯ ) - 1 ( 0 ) .

A *-closed subalgebra of the Smirnov class

We study real Smirnov functions and investigate a certain ∗ * -closed subalgebra of the Smirnov class N + N^+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space H p ∩ H p ¯ H^p \cap \overline {H^p} . This leads to a natural analog of the Riesz projection on a certain quotient space of L p L^p for p ∈ ( 0 , 1 ) p \in (0,1) . We also study a Herglotz-like integral transform for singular measures on the unit circle ∂ D \partial \mathbb {D} .