Abstract. For | X | < 1, A* is the operator defined formally on the Hardy space H2 by (Aif)(z) = -(\-z)-'fj(s)ds, \z\<\.If X = 1, then the usual identification of H2 with I2 takes /I, onto the discrete Cesàro operator. Here we answer questions about boundedness, spectra, unitary equivalence, compactness, and subnormality for the operators Ax.The Cesàro operator C0 acting on the Hilbert space I2 of square-summable complex sequences {ö"}*=0 is defined by C0{an} -{bn} where bn = 1"=0 aJ(n + 1), n = 0,1,2,_This operator was studied extensively in [1] where it was shown, among other things, that C0 is bounded with ||C0|| = 2 and spectrum {z: | 1 -z\< 1}. In [4] it was proved that C0 is a subnormal operator.For 0 <| A |< 1 we define the operator Ax on I2 by Ax{an} = {cn} where c" = 2nJ=0\n-Jaj/(n + 1), n = 0,1,2,...; also define A0 by A0{a"} = {a"/(n + 1)}. Observe that Ax -C0. We identify I2 isometrically with the Hardy space H2by sending {anX?=o onto/(z) = 2^=o u"z". Ax then becomes an operator on H2. A\ can be expressed in closed form as follows. If f(z) = 2^=Qanz", then (Atf)(z) = lf=0 ckzk where ck = ^=k a"X"~k/(n + I). Consider j{ f(s) ds where the path of integration is sufficiently nice. If X -1 and the path consists of two segments, one connecting 1 to 0 and the other connecting 0 to z, we have jx f(s)ds = f¿f(s)ds -/n1 f(s)ds; the last integral exists by the Fejér-Riesz inequality [2, p. 46]. Integrating the Taylor series for/ term-by-term we have f{ f(s) ds = 2?=0a"z"+1/(H + 1) -2?=0a"An+1/(H + 1). Comparing these Taylor coefficients with the Taylor coefficients of (X -z)(A\f) (z) we see that (X-z)(A*J)(z) = -T f(s)ds, \z\
For α ∈ [0, 1] the operator is the operator formally defined on the Hardy space H2 byIf α = 1, then the usual identification of H2 with l2 takes A1 onto the discrete Cesàro operator. Here we see that {Aα: α ∈ [0, 1]} is not arcwise connected, that Re Aα ≥ 0, that Aα is a Hilbert-Schmidt operator if α ∈[0, 1), and that Aα is neither normaloid nor spectraloid if α ∈(0, 1).
First it is shown that the N örlund matrix associated with the sequence of positive integers is a coposinormal operator on ℓ 2 . This fact then turns out to be useful for showing that this operator is also posinormal and hyponormal. In contrast with the analogous weighted mean matrix result [6], the proof of hyponormality is accomplished without resorting to determinants or Sylvester's criterion.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.