The splitting relation for K�the spaces

Krone, Jörg; Vogt, Dietmar

doi:10.1007/bf01215139

Cited by 21 publications

(13 citation statements)

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“…They showed, in particular, that if the spaces satisfy a splitting condition of Apiola type [1], then the existence of an unbounded operator implies the existence of a common basic subspace. Djakov and Ramanujan [2] obtain the same result without the assumption of nuclearity and assuming the weaker splitting condition of Krone and Vogt [5].…”

Section: Proposition 2 We Have (λ(A) λ(B) λ(C)) ∈ Bf If and Only Imentioning

confidence: 53%

“…This is a generalization of Proposition 3.4 from [5], where the case of Köthe spaces was considered (for Köthe spaces the conditions S and S coincide): basically, our proof of Proposition 7 is a generalized direct version of the proof ad absurdum from [5] Proof. Because of complete similarity we consider only the case S. Suppose that (F, E) ∈ S. Then there is a function τ : N → N such that for each T ∈ L(E, F ) the estimate…”

Section: Now We Consider a Factorized Analogue Of The Condition S A mentioning

confidence: 85%

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Factorization of unbounded operators on Köthe spaces

2004

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Abstract.The main result is that the existence of an unbounded continuous linear operator T between Köthe spaces λ(A) and λ(C) which factors through a third Köthe space λ(B) causes the existence of an unbounded continuous quasidiagonal operator from λ(A) into λ(C) factoring through λ(B) as a product of two continuous quasidiagonal operators. This fact is a factorized analogue of the Dragilev theorem [3,6,7, 2] about the quasidiagonal characterization of the relation (λ(A), λ(B)) ∈ B (which means that all continuous linear operators from λ(A) to λ(B) are bounded). The proof is based on the results of [9] where the bounded factorization property BF is characterized in the spirit of Vogt's [10] characterization of B. As an application, it is shown that the existence of an unbounded factorized operator for a triple of Köthe spaces, under some additonal asumptions, causes the existence of a common basic subspace at least for two of the spaces (this is a factorized analogue of the results for pairs [8,2]).

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Section: Proposition 2 We Have (λ(A) λ(B) λ(C)) ∈ Bf If and Only Imentioning

confidence: 53%

Section: Now We Consider a Factorized Analogue Of The Condition S A mentioning

confidence: 85%

Factorization of unbounded operators on Köthe spaces

2004

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“…As the topological duals of k 0 (A) ⊗ ε F and k 0 (A) ⊗ π F coincide, we have to show that ε also equals τ . From L(F, λ 1 (A)) = LB(F, λ 1 (A)) we deduce that the pair (λ 1 (A), F ) fulfills condition (S * 2 ) (see [19,Proposition 3.4]). If λ 1 (A) is countably normed this implies the condition (S 2 ) (see proof of Theorem 2.1).…”

Section: Wwwmn-journalcommentioning

confidence: 99%

“…Using the well-known condition (S * 2 ) of D. Vogt, which plays a central role in the theory of splitting sequences of Fréchet spaces (see [12] and [29]), J. Krone and D. Vogt characterized the barrelledness of the space L b λ 1 (A), λ 1 (B) under the assumption of λ 1 (A) being Schwartz ( [19,Theorem 2.1]). This was generalized by E. Mangino for distinguished λ 1 (A) and an arbitrary Fréchet space instead of λ 1 (B) ([23, Corollary 1]) using fundamental results concerning proj 1 X = 0 (see [8] and [12]).…”

Section: Introductionmentioning

confidence: 99%

A generalization of a theorem of A. Grothendieck

Varol

2007

Mathematische Nachrichten

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In this article we characterize the quasi‐barrelledness of the projective tensor product of a coechelon space of type one k 1(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k 1(A) ·: ℱ → ℒ︁, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor1π (k 1(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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“…If Q is the principal part of P, then P = Q + R with R := P -Q. For [20] together with the representation of ker P ( D ) as a tensor product shows that ker P(D) is not bornological (b) M. LANGENBRUCH has pointed out that Lemma 5.1 can be improved if one uses a sequence space representation for ker P(D) which follows from LANGENBRUCH [21] to show that ker P ( D ) is not bornological.…”

Section: Examplesmentioning

confidence: 99%

Characterization of the Linear Partial Differential Operators with Constant Coefficients Which are Surjective on Non‐quasianalytic Classes of Roumieu Type on ℝ^N

Braun

Meise

Vogt

1994

Mathematische Nachrichten

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After it was observed by DE GIORGI and CATTABRIGA [I41 and PICCININI [31] that the heat operator is not surjective on the space of all real-analytic functions on R3, HORMANDER [ 181 characterized the linear partial differential operators with constant coefficients which are surjective on all real-analytic functions on a given convex open subset of RN. On the other hand, CATTABRIGA [Ill, [12], showed that the heat operator is not surjective on the classical Gevrey classes Tcd1(lR3) for 1 < d < 2. Sufficient conditions for the surjectivity of linear partial differential operators with constant coefficients were given by CATTABRIGA [Ill and ZAMPIERI [34], [35], where the latter used methods of HORMANDER [18].In the present paper we characterize the surjectivity of linear partial differential operators with constant coefficients on the non-quasianalytic classes gcw,(IRN) which extend the Gevrey classes ridl(IRN) for d > 1 (see PETZSCHE and VOGT [30] and BRAUN, MEISE and TAYLOR [S]). More precisely, we show that for a complex polynomial P i n N variables the differential operator P(D): &{wl@2N) 4 E {~~( I R~) is surjective if and only if the zero variety V ( P ) : = ( z E C N : P ( -Z ) = 0)of P satisfies the following equivalent Phragmtn-Lindelof conditions: PL(o):There exist 6 > 0 and A 2 1 such that for each B 2 A and E > 0 there exists C > 0 such that each plurisubharmonic function u on V ( P ) which satisfies u(z) < IIm zI + 6o(z) and u(z) 5 B IIm zI for all z E V also satisfies u(z) I A IIm zI + EO(Z) + C for all z E V .There exist R > 0 and A 2 1 such that for each B 2 A and E > 0 there exists C > 0 such that, for each B E V(P) with 181 2 R, each plurisubharmonic function u on V ( P ) which satisfies u(z) I w(B) and u(z) I B (Im Z( for all z E V ( P ) , )Z -el < o(0) also satisfies u(0) I A /Im 81 + m ( 8 ) + C. PL,(w):') Research partially supported by Deutsche Forschungsgemeinschaft. 2'

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The splitting relation for K�the spaces

Cited by 21 publications

References 7 publications

Factorization of unbounded operators on Köthe spaces

Factorization of unbounded operators on Köthe spaces

A generalization of a theorem of A. Grothendieck

Characterization of the Linear Partial Differential Operators with Constant Coefficients Which are Surjective on Non‐quasianalytic Classes of Roumieu Type on ℝ^N

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The splitting relation for K�the spaces

Cited by 21 publications

References 7 publications

Factorization of unbounded operators on Köthe spaces

Factorization of unbounded operators on Köthe spaces

A generalization of a theorem of A. Grothendieck

Characterization of the Linear Partial Differential Operators with Constant Coefficients Which are Surjective on Non‐quasianalytic Classes of Roumieu Type on ℝN

Contact Info

Product

Resources

About

Characterization of the Linear Partial Differential Operators with Constant Coefficients Which are Surjective on Non‐quasianalytic Classes of Roumieu Type on ℝ^N