Abstract
Let ℋ be a Krein space with fundamental symmetry J. Starting with a canonical block-operator matrix representation of J, we study the regular subspaces of ℋ. We also present block-operator matrix representations of the J-self-adjoint projections for the regular subspaces of ℋ, as well as for the regular complements of the isotropic part in a pseudo-regular subspace of ℋ.
Highlights
An idempotent in B(H) is called a projection
In [4], the authors proved that a closed subspace S of H is pseudoregular if and only if it is the range of a J-normal projection in B(H). ey showed in the same paper that a pseudoregular subspace S admits infinitely many J-normal projections onto it, unless S is regular
In [8], Giribet et al gave a block-operator matrix representation of the fundamental symmetry J depending on a pseudoregular subspace S of H, and from here on, they characterized the J-self-adjoint projections for the regular complements of S0 in S
Summary
For a pseudoregular subspace S of the Krein space H, a block-operator matrix representation of the fundamental symmetry J was obtained with the space decomposition H S0 ⊕ (S ⊖ S0) ⊕ (S⊥ ⊖ J(S0)) ⊕ J(S0) in [8]. Let H be a Krein space with fundamental symmetry J, and let S be a closed subspace of H. en, J has the operator matrix representation:. Noting that J is self-adjoint, J has the operator matrix representation:. With respect to H H1 ⊕ (H4 ⊕ H5) ⊕ H7 ⊕ H8, and by eorem 2, S is regular and the J-self-adjoint projection E onto S has the operator matrix representation:. Noting that JE JE1 − (− JE2), where JE1 and − JE2 are the positive operators in B(H), we see that JE1 and − JE2 are the positive part and the negative part of the self-adjoint operator JE, respectively. JE1 and − JE2, and E1 and E2 are unique
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