J-Self-Adjoint Projections in Krein Spaces

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.

Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

Regular subspaces of skew product diffusions

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space ${E_{1}\times E_{2}}$ and expressed as$X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$where ${X^{i}}$ is a symmetric diffusion on ${E_{i}}$ for ${i=1,2}$, and A is a positive continuous additive functional of ${X^{1}}$. One of our main results indicates that any skew product type regular subspace of X, say$Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$can be characterized as follows: the associated smooth measure of ${\tilde{A}}$ is equal to that of A, and ${Y^{i}}$ corresponds to a regular subspace of ${X^{i}}$ for ${i=1,2}$. Furthermore, we shall make some discussions on rotationally invariant diffusions on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$, which are special skew product diffusions on ${(0,\infty)\times S^{d-1}}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ${\mathbb{R}^{d}}$. More precisely, fix a regular Dirichlet form ${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ${\mathbb{R}^{d}}$. Its part Dirichlet form on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. Let ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. We want to find a regular subspace ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of ${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. If ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.