Spectrum of Convolution Dilation Operators on Weighted L Spaces
نویسندگان
چکیده
R c(x)dx = 1. For any sufficiently large number K the space Lp([−K,K]) of all Lp-functions with support in the interval [−K,K] is an invariant space of Wc,α. It is known that Wc,α restricted to Lp([−K,K]) is a compact operator with eigenvalues α−k, k = 0, 1, . . . , and spectrum {α−k : k = 1, 2, . . .} ∪ {0}, which are independent of c and K. This result is better understood in the context of weighted Lp space, Lw(R) that comprises functions f for which fw belong to Lp(R). We prove that under an oscillation condition on w, Wc,α is a compact operator on Lw(R) if and only if lim|x|→∞ w(x)/w(αx) = 0. Further, Wc,α has exactly the same eigenvalues and spectrum as its restriction to Lp([−K,K]). We also prove that if lim|x|→∞ w(x)/w(αx) = r for some positive constant r, then the spectrum of Wc,α on the space L p w(R) is the closed disc Ds := {λ ∈ C : |λ| ≤ rα1−1/p} in addition to the set {α−k : k = 1, 2, . . .}, and that all nonzero complex numbers with absolute value strictly less than r are eigenvalues of the operator Wc,α on L p w(R). In particular, for w = 1 the results say that the spectrum of Wc,α on Lp(R) is the closed disc with centre at the origin and radius α1−1/p, and that all nonzero complex numbers with absolute value strictly less than 1 are its eigenvalues.
منابع مشابه
Weighted composition operators on weighted Bergman spaces and weighted Bloch spaces
In this paper, we characterize the bonudedness and compactness of weighted composition operators from weighted Bergman spaces to weighted Bloch spaces. Also, we investigate weighted composition operators on weighted Bergman spaces and extend the obtained results in the unit ball of $mathbb{C}^n$.
متن کاملEssential norm estimates of generalized weighted composition operators into weighted type spaces
Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operat...
متن کاملComposition operators acting on weighted Hilbert spaces of analytic functions
In this paper, we considered composition operators on weighted Hilbert spaces of analytic functions and observed that a formula for the essential norm, gives a Hilbert-Schmidt characterization and characterizes the membership in Schatten-class for these operators. Also, closed range composition operators are investigated.
متن کامل