2017
Том 69
№ 6

All Issues

On singular lineals in $\Pi_{\chi}$ spaces

Iokhvidov I. S.

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Abstract

The author consider the Hilbert space $\mathfrak{H}$ in which a $J$-metric $[x, y] = (yx, y)$ is introduced, where $yj$ is the difference of two orthoprojections in $\mathfrak{H}$. The lineal $\mathfrak{L} \subset \mathfrak{H}$ is called definite if the form $[x, x] (x \in \mathfrak{L})$ has a constant sign; the lineal is called singular if the norms $(x, x)^{1/2}$ and $|[ x , x]|^{1/2}$ are nonequivalent. The properties of singular lineals are studied. In particular, it is shown that an arbitrary infinite-dimensional lineal with a positive Hermitian-bilinear metric $[x, y]$, complete with respact to the norm $|x| = [x, x]^{1/2}$ may, preserving the form $[x, y]$, be embeded into space $\Pi_{\chi}$ with an arbitary integer $\chi$ so that it proves to be a singular lineal with a given measure of singularity $m \leq \chi$.

Citation Example: Iokhvidov I. S. On singular lineals in $\Pi_{\chi}$ spaces // Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 300-308.

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