Mean ergodicity and spectrum of the Cesàro operator on weighted $$c_0$$ c 0 spaces

Albanese, Angela A.; Bonet, José; Ricker, Werner J.

doi:10.1007/s11117-015-0385-x

Cited by 11 publications

(40 citation statements)

References 20 publications

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“…It is shown there that there exists a strictly increasing sequence

{false(n (k) false)}_{k \in double-struckN}

double-struckN

with the property: for each

t \in R

we have

\frac{1}{{(n false(k false))}^{t} w false(n (k) false)} \geq k, k > t .

Hence,

t \notin S_{w} false(1 false)

and so

S_{w} false(1 false) = \emptyset

. It is also shown in that

w \notin s

. (ii)If

v, w

are bounded, strictly positive sequences with

w \leq v

, then

\frac{1}{v} \leq \frac{1}{w}

from which it follows that

S_{1} false(w false) \subseteq S_{1} false(v false)

. Hence,

inf S_{1} false(v false) \leq inf S_{1} false(w false)

.…”

Section: Spectrum Of Cfalse(1wfalse)mentioning

confidence: 93%

“…Proof Clearly,

(I - {sans-serifC}^{(1, w)}) false(X_{1} (w) false) \subseteq (I - {sans-serifC}^{(1, w)}) false(ℓ_{1} (w) false)

. To verify the reverse inclusion, we proceed as in the proof of [, Lemma 4.5]. First observe, via , that for each

x \in ℓ_{1} false(w false)

we have

(I - {sans-serifC}^{(1, w)}) x = (0, x_{2} - \frac{x_{1} + x_{2}}{2}, x_{3} - \frac{x_{1} + x_{2} + x_{3}}{3}, \dots),

and, in particular, for each

y \in X_{1} false(w false)

that

true(I - C^{false(1, w false)} true) y = (0, \frac{y_{2}}{2}, y_{3} - \frac{y_{2} + y_{3}}{3}, y_{4} - \frac{y_{2} + y_{3} + y_{4}}{4}, \dots) .

Fix

x \in ℓ_{1} false(w false)

.…”

Section: Iterates Of Cfalse(1wfalse) and Mean Ergodicitymentioning

confidence: 98%

“…So, for

β \geq 1

the compactness of

C^{false(1, w_{β} false)}

does follow from Proposition . (vi)Fix

γ > 1

and set

w_{γ} false(n false) : = e^{- {prefixlog}^{γ} false(n false)}

for

n \in N

. It is shown in [, Remark 2.10(ii)] that

{trueprefixlim}_{n \to \infty} \frac{w_{γ} false(n + 1 false)}{w_{γ} false(n false)} = 1

and so Proposition is not applicable. However,

\begin{matrix} true \\ right A_{n} : = \frac{1}{w_{γ} false(n false)} \sum_{m = n}^{\infty} \frac{w_{γ} false(m false)}{m} = e^{{prefixlog}^{γ} false(n false)} \sum_{m = n}^{\infty} \frac{e^{- {log}^{γ} (m)}}{m} \leq e^{{prefixlog}^{γ} false(n false)} \int_{n - 1}^{\infty} \frac{e^{- {log}^{γ} (x)}}{x} d x, n \geq 2, \end{matrix}

because

x \mapsto \frac{e^{- {log}^{γ} (v)}}{x} = \frac{1}{x {prefixlog}^{γ} false(x false)}

is decreasing in (1, ∞).…”

Section: Continuity and Compactness Of Sans-serifc In ℓ1false(wfalse)mentioning

confidence: 99%

“…Remark (i)The converse of Proposition (iii) is not valid. Indeed, let

w = {(w false(n false))}_{n \in N}

be the strictly positive weight with

w ↓ 0

as given in [, Remark 3.2]. It is shown there that there exists a strictly increasing sequence

{false(n (k) false)}_{k \in double-struckN}

double-struckN

with the property: for each

t \in R

we have

\frac{1}{{(n false(k false))}^{t} w false(n (k) false)} \geq k, k > t .

Hence,

t \notin S_{w} false(1 false)

and so

S_{w} false(1 false) = \emptyset

.…”

Section: Spectrum Of Cfalse(1wfalse)mentioning

confidence: 99%

“…Amongst the classical Banach spaces

X \subseteq C^{double-struckN}

where answers are known we mention

ℓ_{p}

(

1 < p < \infty

), , and c 0 , , both c ,

ℓ_{\infty}

, , as well as

c e s_{p}

p \in {0} \cup (1, \infty)

, , the spaces of bounded variation

b v_{0}

, , and

b v_{p}

1 \leq p < \infty

, , and the Bachelis spaces

N^{p}

1 < p < \infty

, . For

sans-serifC

acting in the weighted Banach spaces

ℓ_{p} false(w false)

1 < p < \infty

, and

c_{0} false(w false)

we refer to . There is no claim that this list of spaces (and references) is complete; see also .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Cesàro operator in weighted ℓ₁ spaces

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“…It is shown there that there exists a strictly increasing sequence

{false(n (k) false)}_{k \in double-struckN}

double-struckN

with the property: for each

t \in R

we have

\frac{1}{{(n false(k false))}^{t} w false(n (k) false)} \geq k, k > t .

Hence,

t \notin S_{w} false(1 false)

and so

S_{w} false(1 false) = \emptyset

. It is also shown in that

w \notin s

. (ii)If

v, w

are bounded, strictly positive sequences with

w \leq v

, then

\frac{1}{v} \leq \frac{1}{w}

from which it follows that

S_{1} false(w false) \subseteq S_{1} false(v false)

. Hence,

inf S_{1} false(v false) \leq inf S_{1} false(w false)

.…”

Section: Spectrum Of Cfalse(1wfalse)mentioning

confidence: 93%

“…Proof Clearly,

(I - {sans-serifC}^{(1, w)}) false(X_{1} (w) false) \subseteq (I - {sans-serifC}^{(1, w)}) false(ℓ_{1} (w) false)

. To verify the reverse inclusion, we proceed as in the proof of [, Lemma 4.5]. First observe, via , that for each

x \in ℓ_{1} false(w false)

we have

(I - {sans-serifC}^{(1, w)}) x = (0, x_{2} - \frac{x_{1} + x_{2}}{2}, x_{3} - \frac{x_{1} + x_{2} + x_{3}}{3}, \dots),

and, in particular, for each

y \in X_{1} false(w false)

that

true(I - C^{false(1, w false)} true) y = (0, \frac{y_{2}}{2}, y_{3} - \frac{y_{2} + y_{3}}{3}, y_{4} - \frac{y_{2} + y_{3} + y_{4}}{4}, \dots) .

Fix

x \in ℓ_{1} false(w false)

.…”

Section: Iterates Of Cfalse(1wfalse) and Mean Ergodicitymentioning

confidence: 98%

“…So, for

β \geq 1

the compactness of

C^{false(1, w_{β} false)}

does follow from Proposition . (vi)Fix

γ > 1

and set

w_{γ} false(n false) : = e^{- {prefixlog}^{γ} false(n false)}

for

n \in N

. It is shown in [, Remark 2.10(ii)] that

{trueprefixlim}_{n \to \infty} \frac{w_{γ} false(n + 1 false)}{w_{γ} false(n false)} = 1

and so Proposition is not applicable. However,

\begin{matrix} true \\ right A_{n} : = \frac{1}{w_{γ} false(n false)} \sum_{m = n}^{\infty} \frac{w_{γ} false(m false)}{m} = e^{{prefixlog}^{γ} false(n false)} \sum_{m = n}^{\infty} \frac{e^{- {log}^{γ} (m)}}{m} \leq e^{{prefixlog}^{γ} false(n false)} \int_{n - 1}^{\infty} \frac{e^{- {log}^{γ} (x)}}{x} d x, n \geq 2, \end{matrix}

because

x \mapsto \frac{e^{- {log}^{γ} (v)}}{x} = \frac{1}{x {prefixlog}^{γ} false(x false)}

is decreasing in (1, ∞).…”

Section: Continuity and Compactness Of Sans-serifc In ℓ1false(wfalse)mentioning

confidence: 99%

“…Remark (i)The converse of Proposition (iii) is not valid. Indeed, let

w = {(w false(n false))}_{n \in N}

be the strictly positive weight with

w ↓ 0

as given in [, Remark 3.2]. It is shown there that there exists a strictly increasing sequence

{false(n (k) false)}_{k \in double-struckN}

double-struckN

with the property: for each

t \in R

we have

\frac{1}{{(n false(k false))}^{t} w false(n (k) false)} \geq k, k > t .

Hence,

t \notin S_{w} false(1 false)

and so

S_{w} false(1 false) = \emptyset

.…”

Section: Spectrum Of Cfalse(1wfalse)mentioning

confidence: 99%

“…Amongst the classical Banach spaces

X \subseteq C^{double-struckN}

where answers are known we mention

ℓ_{p}

(

1 < p < \infty

), , and c 0 , , both c ,

ℓ_{\infty}

, , as well as

c e s_{p}

p \in {0} \cup (1, \infty)

, , the spaces of bounded variation

b v_{0}

, , and

b v_{p}

1 \leq p < \infty

, , and the Bachelis spaces

N^{p}

1 < p < \infty

, . For

sans-serifC

acting in the weighted Banach spaces

ℓ_{p} false(w false)

1 < p < \infty

, and

c_{0} false(w false)

we refer to . There is no claim that this list of spaces (and references) is complete; see also .…”

Section: Introductionmentioning

confidence: 99%

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Mean ergodicity and spectrum of the Cesàro operator on weighted $$c_0$$ c 0 spaces

Cited by 11 publications

References 20 publications

The Cesàro operator in weighted ℓ₁ spaces

The Cesàro operator in weighted ℓ₁ spaces

On the spectra of multidimensional normal discrete Hausdorff operators

The Cesàro Operator on Duals of Smooth Sequence Spaces of Infinite Type

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Mean ergodicity and spectrum of the Cesàro operator on weighted $$c_0$$ c 0 spaces

Cited by 11 publications

References 20 publications

The Cesàro operator in weighted ℓ1 spaces

The Cesàro operator in weighted ℓ1 spaces

On the spectra of multidimensional normal discrete Hausdorff operators

The Cesàro Operator on Duals of Smooth Sequence Spaces of Infinite Type

Contact Info

Product

Resources

About

The Cesàro operator in weighted ℓ₁ spaces

The Cesàro operator in weighted ℓ₁ spaces