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951.
Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily. 相似文献
952.
Let X and Y be superreflexive complex Banach spaces and let B(X) and B(Y) be the Banach algebras of all bounded linear operators on X and Y, respectively. If a bijective linear map Φ:B(X)→B(Y) almost preserves the spectra, then it is almost multiplicative or anti-multiplicative. Furthermore, in the case where X=Y is a separable complex Hilbert space, such a map is a small perturbation of an automorphism or an anti-automorphism. 相似文献
953.
We show that the absolute numerical index of the space Lp(μ) is (where ). In other words, we prove that
954.
We study in dimension d?2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrödinger operators with a radially symmetric potential falling off like −γr−2, γ>0. We consider angular momentum sectors, labelled by l=0,1,…, for which γ>2(l+d/2−1). In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift. 相似文献
955.
If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C?-subalgebra of τ generated by Toeplitz operators with symbols in H∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M1 of trivial Gleason parts. 相似文献
956.
Let T be a Cowen-Douglas operator. In this paper, we study the von Neumann algebra V?(T) consisting of operators commuting with both T and T? from a geometric viewpoint. We identify operators in V?(T) with connection-preserving bundle maps on E(T), the holomorphic Hermitian vector bundle associated to T. By studying such bundle maps, the structure of V?(T) as well as information on reducing subspaces of T can be determined. 相似文献
957.
The composition operators on H2 whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH2 of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H2, but there is a similarity between them. 相似文献
958.
We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role. 相似文献
959.
Vadim Mogilevskii 《Journal of Functional Analysis》2011,261(7):1955-1968
Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator. 相似文献
960.
J.M. Calabuig E.A. Sánchez-Pérez 《Journal of Mathematical Analysis and Applications》2011,373(1):316-321
Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L∞(μ)) if and only if the equality T(g〈f,x∗〉x)=g〈T(f),x∗〉x holds for every g∈L∞(μ), f∈E(X), x∈X and x∗∈X∗. 相似文献