Extended eigenvalues for Cesaro operators

作者：Lacruz Miguel^*; Leon Saavedra Fernando; Petrovic Srdjan; Zabeti Omid

来源：Journal of Mathematical Analysis and Applications, 2015, 429(2): 623-657.

DOI：10.1016/j.jmaa.2015.04.028

• Extended eigenvalue
• Cesaro operator
• Euler operator
• Rich point spectrum
• Bilateral weighted shift
• Analytic Toeplitz operator

A complex scalar A is said to be an extended eigenvalue of abounded linear operator T on a complex Banach space if there is a non-zero operator X such that TX = lambda XT. Such an operator X is called an extended eigenoperator of T corresponding to the extended eigenvalue A. The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesaro operator C-0, the finite continuous Cesaro operator C-1 and the infinite continuous Cesaro operator Coo defined on the complex Banach spaces l(p), L-p[0, 1] and L-p[0, infinity) for 1 < p < infinity by the expressions
(C(0)f)(n): = 1/n+1 Sigma(n)(k=0)f(k),
(C(1)f)(x): = 1/x integral(x)(0) f(t)dt,
(C(infinity)f)(x): = 1/x integral(x)(0) f(t)dt.
It is shown that the set of extended eigenvalues for C-0 is the interval [1, infinity), for C-1 it is the interval (0,1], and for C-infinity it reduces to the singleton {1}.

• 出版日期2015-9-15