We consider the convolution dilation operator Wc,αf (x) = α R c(αx − y)f (y)dy, f ∈ L p (R), where α is a real number strictly larger than 1, and c is a compactly supported integrable kernel with R c(x)dx = 1. For any sufficiently large number K the space L p ([−K, K]) of all L p-functions with support in the interval [−K, K] is an invariant space of Wc,α. It is known that Wc,α restricted to L p ([−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 L p space, L p w (R) that comprises functions f for which f w belong to L p (R). We prove that under an oscillation condition on w, Wc,α is a compact operator on L p w (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 L p ([−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| ≤ rα 1−1/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 L p (R) is the closed disc with centre at the origin and radius α 1−1/p , and that all nonzero complex numbers with absolute value strictly less than 1 are its eigenvalues.
