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The conjugate of a product of linear relations

Jacob J. Jaftha (2006)

Commentationes Mathematicae Universitatis Carolinae

Let X , Y and Z be normed linear spaces with T ( X → Y ) and S ( Y → Z ) linear relations, i.e. setvalued maps. We seek necessary and sufficient conditions that would ensure that ( S T ) ' = T ' S ' . First, we cast the concepts of relative boundedness and co-continuity in the set valued case and establish a duality. This duality turns out to be similar to the one that exists for densely defined linear operators and is then used to establish the necessary and sufficient conditions. These conditions are similar to those for the single...

The continuity of Lie homomorphisms

Bernard Aupetit, Martin Mathieu (2000)

Studia Mathematica

We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.

The continuity of pseudo-differential operators on weighted local Hardy spaces

Ming-Yi Lee, Chin-Cheng Lin, Ying-Chieh Lin (2010)

Studia Mathematica

We first show that a linear operator which is bounded on L ² w with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space h ¹ w if and only if this operator is uniformly bounded on all h ¹ w -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to h ¹ w .

The continuity of superposition operators on some sequence spaces defined by moduli

Enno Kolk, Annemai Raidjõe (2007)

Czechoslovak Mathematical Journal

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = ( ϕ k ) let λ ( Φ ) = { x = ( x k ) ( ϕ k ( | x k | ) ) ∈ λ } . Given another sequence of modulus functions Ψ = ( ψ k ) , we characterize the continuity of the superposition operators P f from λ ( Φ ) into μ ( Ψ ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕ k ( k ∈ ...

The cubic Szegő equation

Patrick Gérard, Sandrine Grellier (2010)

Annales scientifiques de l'École Normale Supérieure

We consider the following Hamiltonian equation on the L 2 Hardy space on the circle, i ∂ t u = Π ( | u | 2 u ) , where Π is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...

The decomposability of operators relative to two subspaces

A. Katavolos, M. Lambrou, W. Longstaff (1993)

Studia Mathematica

Let M and N be nonzero subspaces of a Hilbert space H satisfying M ∩ N = {0} and M ∨ N = H and let T ∈ ℬ(H). Consider the question: If T leaves each of M and N invariant, respectively, intertwines M and N, does T decompose as a sum of two operators with the same property and each of which, in addition, annihilates one of the subspaces? If the angle between M and N is positive the answer is affirmative. If the angle is zero, the answer is still affirmative for finite rank operators but there are...

The density of states of a local almost periodic operator in ℝ ν

Andrzej Krupa (2003)

Studia Mathematica

We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on H m ( ℝ ν ) . The support of the density coincides with the spectrum of the operator in L ² ( ℝ ν ) .

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