Cesàro Operators

Brown, Arlen; Halmos, Paul R.; Shields, Allen

doi:10.1007/978-1-4613-8208-9_21

Cited by 70 publications

(120 citation statements)

References 6 publications

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“…The discrete Cesàro operator KQ acting on the Hubert space / of square summable complex sequences is defined by x Jo The basic properties of these operators were first discussed by Brown, Halmos, and Shields in [4]. It was shown in [4,Theorems 3,4,and 5] that KQ is a hyponormal operator, 1 -K*x is the unilateral shift operator on L (0, 1 ), and 1 -K*x is the bilateral shift operator on L (0, oo).…”

Section: Introductionmentioning

confidence: 99%

“…It was shown in [4,Theorems 3,4,and 5] that KQ is a hyponormal operator, 1 -K*x is the unilateral shift operator on L (0, 1 ), and 1 -K*x is the bilateral shift operator on L (0, oo). In [12,Theorem 4] it was shown that Kr, is in fact subnormal.…”

Section: Introductionmentioning

confidence: 99%

“…In fact it has been shown in [4] that if T = 1 -KQ , then 1 -TT* is the diagonal operator defined by (l-TT*)en = len for n = 1, 2.…”

Section: Introductionmentioning

confidence: 99%

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Some Trace Class Commutators of Trace Zero

Kittaneh¹

1991

Proceedings of the American Mathematical Society

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Abstract.It is shown that if T is an operator on a separable complex Hubert space and X is a Hilbert-Schmidt operator such that TX -XT is a trace class operator, then the trace of TX -XT is zero provided one of the two conditions holds: (a) T is normal; (b) T" is normal for some integer n > 2 and T'T -TT* is a trace class operator. Related results involving essentially unitary operators and Cesàro operators are also given.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some Trace Class Commutators of Trace Zero

Kittaneh¹

1991

Proceedings of the American Mathematical Society

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“…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.…”

mentioning

confidence: 99%

“…The diagonal operator A0 is compact and Hermitian and has spectrum {i}^°=1 U {0}. The Cesàro operator Ax was studied in [1] and [4]. We shall now restrict ourselves to 0 < a < 1.…”

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confidence: 99%

Discrete generalized Cesàro operators

Rhaly¹

1982

Proc. Amer. Math. Soc.

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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\

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Orbits of Cesàro type operators

León-Saavedra

Lerena

Seoane‐Sepúlveda

2009

Mathematische Nachrichten

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A bounded linear operator T on a Banach space X is called hypercyclic if there exists a vector x ∈ X such that its orbit, {T n x}, is dense in X. In this paper we show hypercyclic properties of the orbits of the Cesàro operator defined on different spaces. For instance, we show that the Cesàro operator defined onMoreover, it is chaotic and it has supercyclic subspaces. On the other hand, the Cesàro operator defined on other spaces of functions behave differently. Motivated by this, we study weighted Cesàro operators and different degrees of hypercyclicity are obtained. The proofs are based on the classical Müntz-Szász theorem. We also propose problems and give new directions.

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Cesàro Operators

Cited by 70 publications

References 6 publications

Some Trace Class Commutators of Trace Zero

Some Trace Class Commutators of Trace Zero

Discrete generalized Cesàro operators

Orbits of Cesàro type operators

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