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Proper subspaces and compatibility

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Andruchow E., Chiumiento E., Di Iorio y Lucero M.

Studia Mathematica

2015

tom 231

nr 3

195-218

Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.

Ograniczanie wyników

1 Studia Mathematica

1 Andruchow E.

1 Chiumiento E.

1 Di Iorio y Lucero M.

1 2015

Wyniki wyszukiwania

Proper subspaces and compatibility