For linear operators, if 1 ≤ p ≤ q > ∞ , 1\leq p\leq q>\infty , then every absolutely p p -summing operator is also absolutely q q -summing. On the other hand, it is well known that for n ≥ 2 , n\geq 2, there are no general “inclusion theorems”for absolutely summing n n -linear mappings or n n -homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely p p -summing and absolutely q q -summing linear operators coincide, and prove that for 1 ≤ p ≤ q ≤ 2 1\leq p\leq q\leq 2 and n ≥ 2 n\geq 2 , we have inclusion theorems for absolutely summing n n -linear mappings/ n n -homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as p p increases, the classes of absolutely p p -summing mappings becomes smaller.

We explore a condition under which the ideal of polynomials generated by an ideal of multilinear mappings between Banach spaces is a global holomorphy type. After some examples and applications, this condition is studied in its own right. A final section provides applications to the ideals formed by multilinear mappings and polynomials which are absolutely (p; q)-summing at every point.

We construct a factorization of certain multilinear mappings through linear operators belonging to closed, injective operator ideals using interpolation technique. An extension of the duality theorem for interpolation spaces is also obtained.

. For several applications it is very useful to classify the linear or non-linear mappings by their summability properties. Absolutely summing operators and polynomials are prominent and classical examples of such setting. Here we are interested in the larger class of almost summing polynomials and we investigate their connections to other related notions of summability.

Approximation numbers of linear operators are a very useful tool in order to understand the structure and the numerical behaviour of the operators. In this paper, this concept is extended to polynomials on Banach spaces and the approximation numbers of diagonal polynomials are estimated. As a main tool the rank of polynomials as a graduation of finite type polynomials is introduced and studied.

Der Begriff der Ableitung einer Funktion kann sowohl geometrisch als Tangentenanstieg als auch analytisch als Änderungsrate der Funktion eingeführt werden. Beide Konzepte ergänzen sich hervorragend. Der analytische Zugang ist jedoch wesentlich leistungsfähiger: Er liefert quantitative Ergebnisse, ermöglicht Existenzuntersuchungen und ist verallgemeinerungsfähig. Zur Vorbereitung definieren wir noch:

We introduce a graduation of unconditionally summing operators and polynomials in two different ways: One way consists in the scaling of the unconditionally p-summability, the other in the scaling of the degree of compactness at the point zero. As a main result it is shown that unconditionally p-summability coincides with the weakly p-compactness for certain parameters p.

Let D be any dense domain in a Hilbert space and let \mathcal L^+ (D) be the maximaximal Op*-algebra of (possibly unbounded) linear operators. In this paper the uniform topology \tau_D on \mathcal L^+ (D) is investigated for the case where D is a DF-space with respect to the graph topology. As a main result, a characterization of the bounded subsets of D and of the topology \tau_D by strongly bounded selfadjoint operators is given. Especially, each bounded subset of D is contained in some bounded ellipsoid. This is applied to approximate the operators in \mathcal L^+ (D) by bounded ones.

This paper deals with the factorization of linear operators mapping (F)-spaces into (DF)-spaces through Banach spaces and through operators of given operator ideals. Roughly speaking, we answer the question of, to what extend global properties of such operators are determined by their behaviour on the hounded subsets. The results are used to characterize the geometric structure of the neighbourhoods of zero in (F)-spaces by the geometry of their bounded sub-sets. Moreover, they allow further insights into the theory of nuclear (F)-spaces.