A characterization of (∞,p,q)-integral operators

Heinrich, Stefan; Pietsch, Albrecht

doi:10.1002/mana.19790890117

Cited by 4 publications

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“…Given B E 8 and any family (zi) with zi E B (i E I ) , (12) implies that lim9 pi(Qisi) < 00, hence the family (Qixi) defines an element of This shows that For u E M l we have a, = e(Ian), h, E eq ( E " / 9 J ( y i ) a , a n ) = ((zi)g, a Step 4 (Proof of the claim). We shall use the local duality 3.3 for the product (Ei)a.…”

Section: (Y E N I E I )mentioning

confidence: 99%

Ultrapowers of Locally Convex Spaces and Applications, I

Heinrich¹

1984

Mathematische Nachrichten

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The author is grateful to C. W. HENSON, A. P E L C Z Y~~S~ and A. PIETSCH for discussions on the subject of this paper. PreliminariesLet E be a locally convex vector space. B(E) denotes the set of all absolutely convex, closed bounded sets, U(E) stands for the set of all closed absolutely convex neighbourhoods of zero, and '$(E) is the set of continuous seminorms. Since there is a one-to-one correspondence between U(E) and '$(E) via the gauge functionals, we also write p u for the seminorm corresponding to U E U ( E ) . Now let U E U(E) and B E % ( E ) be given. By Eu we denote the completion of the quotient space E / { x E E : pu(x) = O } , endowed with the norm pu. The space EB is defined as EB = u {IB : I > O } , equipped with the gauge norm T~( X ) = inf { I > 0 : x E AB), for x E EB. As usual, E' denotes

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Section: (Y E N I E I )mentioning

confidence: 99%

Ultrapowers of Locally Convex Spaces and Applications, I

Heinrich¹

1984

Mathematische Nachrichten

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Obituary of Albrecht Pietsch

Heinrich,

Hinrichs,

Kühn

2024

Mathematische Nachrichten

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The closed ideal of (c o ,p,q)-summing operators

Hinrichs

Pietsch

2000

Integr equ oper theory

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For 1/p+ 1/q<_l, we study the closed ideal ~3r formed by the (co, p, q)-summing operators. It turns out that T : X--+Y does not belong to 913~o,p,q if and only if it factors the map Id : lp. --+lq. By localization, we get the ideal es that consists of those operators T for *-Co ,p,q ~uper which all ultrapowers T u are contained in ~3Co,p,q. Operators in the complement of ~o,p,q are characterized by the property that they factor the maps Id : l~, -+ l~ uniformly. Our main tools are ideal norms. NotationThroughout, we adopt the standard notation used in [P-W]. In particular, t2(X, Y) stands for the set of all (bounded linear) operators acting from a Banach space X into a Banach space Y, and we let lip + 1/p* = 1.

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A characterization of (∞,p,q)-integral operators

Cited by 4 publications

References 1 publication

Ultrapowers of Locally Convex Spaces and Applications, I

Ultrapowers of Locally Convex Spaces and Applications, I

Obituary of Albrecht Pietsch

The closed ideal of (c o ,p,q)-summing operators

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