Banach actions preserving unconditional convergence

DOI10.3390/AXIOMS11010013arXiv2111.14253WikidataQ114582209 ScholiaQ114582209MaRDI QIDQ6384159

Publication date: 28 November 2021

Abstract: Let

A, X, Y

be Banach spaces and

A i m e s X o Y

,

(a, x) m a p s t o a x

, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence

(a_{n})_{n i n o m e g a}

in

A

and unconditionally convergent series

s u m_{n i n o m e g a} x_{n}

in

X

the series

s u m_{n i n o m e g a} a_{n} x_{n}

is unconditionally convergent. We prove that a Banach action

A i m e s X o Y

preserves unconditional convergence if and only if for any linear functional

y^{*} i n Y^{*}

the operator

D_{y^{*}} : X o A^{*}

,

D_{y^{*}} (x) (a) = y^{*} (a x)

, is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from

e l l_{1}

to

e l l_{2}

, we prove that a Banach action

A i m e s X o Y

preserves unconditional convergence if

A

is a Hilbert space possessing an orthonormal basis

(e_{n})_{n i n o m e g a}

such that for every

x i n X

the series

s u m_{n i n o m e g a} e_{n} x

is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers

p, q, r i n [1, i n f t y]

with

f r a c 1 r l e f r a c 1 p + f r a c 1 q

, the coordinatewise multplication

e l l_{p} i m e s e l l_{q} o e l l_{r}

preserves unconditional convergence if and only if one of the following conditions holds: (i)

p l e 2

and

q l e r

, (ii)

2 < p < q l e r

, (iii)

2 < p = q < r

, (iv)

r = i n f t y

, (v)

2 l e q < p l e r

, (vi)

q < 2 < p

and

f r a c 1 p + f r a c 1 q g e f r a c 1 r + f r a c 12

.

Mathematics Subject Classification ID

Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces (46B15) Banach sequence spaces (46B45)

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