Abstract
The random (or stochastic) approximation‐solvability, based on a projection scheme, of linear random operator equations involving the theory of the numerical range of a bounded linear random operator is considered. The obtained results generalize results with regard to the deterministic approximation‐solvability of linear operator equations using the Galerkin convergence method.
Highlights
The theory of random operator equations originated from a desire to develop deterministic operator equations that were more application-oriented, with a special desire to deal with various natural systems in applied mathematics, since the behavior of natural systems is governed by chance
As attempts were made by many scientists and mathematicians to develop and unify the theory of random equations employing concepts and methods of probability theory and functional analysis, the Prague School of probabilists under Spacek initiated a systematic study using probabilistic operator equations as models for various systems
Our aim has been to apply the theory of a random numerical range to the random approximation-solvability of linear random operator equations
Summary
The theory of random operator equations originated from a desire to develop deterministic operator equations that were more application-oriented, with a special desire to deal with various natural systems in applied mathematics, since the behavior of natural systems is governed by chance. Definition 1.1" Let T:W x XX be a linear random operator. Generalized nverses of hnear operators seem to have nice applications in analysis, statistics, prediction and control theory As most of these applications are related to the least-squares property that the generalized inverses possess in Hilbert spaces, T + is characterized by the following extremal property. Definition 1.3: (The least-squares solution): Let T be a linear operator form a Hilbert space X into another Hilbert space Y. Let us recall the random version of a best-approximate solution. Let T be a.s. a bounded random linear operator. Lemma 1.5: [15, Theorem 18El: Let A:X---X be a continuous linear operator on a Hilbert space X over the field K (real or complex). For each given f in X, the operator equation Au f (u in X) has a unique solution
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