Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors
| Autor: | Milton Rosenberg |
|---|---|
| Rok vydání: | 1978 |
| Předmět: |
Statistics and Probability
Banach space Space (mathematics) p × qL2.xF Bounded operator bounded operator-forming vectors Wold decomposition Combinatorics symbols.namesake q × q spectral measure q-variate stationary process associated discrete parameter process p × q spectral integral p × q stochastic integral p × q correlation operator Mathematics Gramian matrix contraction operator semigroup q × q positive-definite function Discrete mathematics Numerical Analysis p × q transition operator Operator (physics) Hilbert space HFq q-variate wide-sense Markov process Linear map semigroup Bounded function symbols Statistics Probability and Uncertainty nondeterministic process |
| Zdroj: | Journal of Multivariate Analysis. 8:295-316 |
| ISSN: | 0047-259X |
| DOI: | 10.1016/0047-259x(78)90081-7 |
| Popis: | Let p, q be arbitrary parameter sets, and let H be a Hilbert space. We say that x = (xi)iϵq, xi ϵ H, is a bounded operator-forming vector (ϵHFq) if the Gram matrix 〈x, x〉 = [(xi, xj)]iϵq,jϵq is the matrix of a bounded (necessarily ≥ 0) operator on lq2, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from lq2 to lp2. Then exists a linear operator Ǎ from (the Banach space) HFq to HFp on D(A) = {x:x ϵ HFq, A〈x, x〉12 is p × q bounded on lq2} such that y = Ǎx satisfies yj ϵ σ(x) = {space spanned by the xi}, 〈y, x〉 = A〈x, x〉 and 〈y, y〉 = A〈x, x〉12(A〈x, x〉12)∗. This is a generalization of our earlier [J. Multivariate Anal. 4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes. |
| Databáze: | OpenAIRE |
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