Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

Noncommutative Figà-Talamanca–Herz Algebras for Schur Multipliers

In this work, we introduce a noncommutative analogue of the Figà-Talamanca–Herz algebra A _p (G) on the natural predual of the operator space \frakM_p,cb{\frak{M}_{p,cb}} of completely bounded Schur multipliers on the Schatten space S _p . We determine the isometric Schur multipliers and prove that the space \frakM_p{\frak{M}_{p}} of bounded Schur multipliers on the Schatten space S _p is the closure in the weak operator topology of the span of isometric multipliers.     

Fredholmness of multipliers on Hardy–Sobolev spaces

This paper discusses the Fredholmness of multipliers on Hardy–Sobolev Spaces and obtains an index formula for the multipliers with some special symbols. Our results show that Hardy–Sobolev spaces have richer properties than classical holomorphic function spaces, and the behavior of the operators on these spaces is complex. Some methods of Hardy or Bergman spaces fail in the case of the Hardy–Sobolev space.     

Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t)dt+dW _H (t), t0, where A is the generator of a C ₀-semigroup S on a separable real Banach space E and W _H (t) _t0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S _H .     

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.     

Some Computations of Frobenius–Schur Indicators of the Regular Representations of Hopf Algebras

We study Frobenius–Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational results, we formulate the theorem of Frobenius for semisimple Hopf algebras and give some partial results on this problem.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

On the theory of Besov–Herz spaces and Euler equations

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\). We show similar results involving the norm distortion ‖T‖‖T ^?1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.     

A Note on Semigroups of Linear Operators

It is well known[l]that a closed linear operator A with dense domain in(an ordered)Banach space E senerates a C_O—semigroup if and only if the resolvent set p(A)contains(ω,+∞),andsup{‖(λ-ω)~nR(λ,A)~n‖:λ>ω,n∈N}<+∞(1)for someω.Sometimes,the condition(l)is not easy to verify.However,for the socalled resolvent positive operators(that is for someω,p(A)(?)(ω,+∞)and for allλ>ω,R(λ,A)=(λ-A)~1≥0),in[2],there are some easy-verified conditions.Here in thisnote,we establish another easy-verified condition.Let E_+be the positive cone of E,and     

A geometric Schur–Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur–Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.     

Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi–Trudi formulas and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulas we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.     

Spectral multipliers on Heisenberg–Reiter and related groups

Annali di Matematica Pura ed Applicata (1923 -) - Let $$L$$ be a homogeneous sublaplacian on a $$2$$ -step stratified Lie group $$G$$ of topological dimension $$d$$ and homogeneous dimension $$Q$$...     

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this p...