(such an extension exists by virtue of Zorn’s lemma). Under hypotheses as above, Equation (4) has a unique solution The following theorem asserts the existence and uniqueness of generalized solution of (4). Under the above hypotheses, there exist the dual mappingsīeing strictly monotone, single-valued, homogeneous, hemi-continuous and such that Is also a hemi-continuous monotone operator from X into We now deal with the stable method of computing values of the operator A at Is open or everywhere dense in X, or if A is maximal monotone, then a generalized solutionĬoincides with the corresponding solution We note that, if A is hemi-continuous and If A is an arbitrary monotone operator, we follow and understand a solution of (1) to be an elementĪ generalized solution of Equation (1). If A is a maximal monotone (possibly multi-valued). In - a class of monotone operators was singled out and, as an approximate method, the operator-regularization method was used.Īs it is known, a solution of (1) is understood to be an element These problems are important objects of investigation in the theory unstable problems. We consider the following three problemsģ) To compute values of the operator A at (possibly multi-valued) and y is a given element in Is a hemi-continuous monotone operator from X into Let X be a real strictly convex reflexive Banach space with the dual 143 Putnam's theorem, 54 quotient Banach space, 184, 185 C-algebra. The Stable Method of Computing Values of Hemi-Continuous Monotone Operators 22 on positive self-adjoint operators, 158 phase of a bounded operator. The approximate values of the operator A atģ.
In a similar way as above, the everywhere defined inverse The eigenvectors are complete in the sense that eigenvectors form some sort of. Because of the uniqueness of decomposition (7), x is uniquely determined by z, and so the everywhere defined inverse lar(possibly unbounded) self-adjoint operator defined on the Hilbert space.
, we have the uniquely determined decomposition Is a closed densely defined linear operator thenĪre complementary orthogonal subspaces of the Hilbert space If A : H H is a bounded linear map, its adjoint A : H. The following lemma will be used in the proof of Theorem 2.2. From now on, we restrict our attention to linear operators from a Hilbert space. To further simplify the presentation, we introduce the functions Palmer Author content Content may be subject to copyright. Palmer University of Oregon Content uploaded by Theodore W. To establish the convergence of (3), it will be convenient to reformulate (3) asĪre known to be bounded everywhere defined linear operators and Unbounded normal operators on Banach spaces Authors: Theodore W. The minimization problem (1) has a unique solution Is also a closed densely defined unbounded linear operator from X into Y with domainįirst, we define the regularization functional Is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y with domain The Stable Method of Computing Values of Closed Densely Defined Unbounded Linear Operators In this paper we shall be concerned with the construction of a stable method of computing values of the operator A for the perturbations (2).Ģ. This leads to a self-adjoint extension of an unbounded operator, which is known as the Friedrichs extension. Until now, this problem is still an open problem. We should approximate values of A when we are given the approximations We now assume that both the operator A and In the another case, where A is a monotone operator from a real strictly convex reflexive Banach space X into its dual
Moreover, the order of convergence results for Morozov has studied a stable method for approximating the value In the case, where A is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y, V. Unbounded operators on Hilbert spaces and their spectral theory Adjoint of a densely de ned operator Self-adjointess Spectrum of unbounded operators on Hilbert spaces Basics Example: 1 For any space X, the bounded linear operators B(X), form a Banach algebra with identity 1 X. , we can see that the values of the operator A may not even be defined on the elements Therefore, the problem of computing values of an operator in the considered case is unstable. , where X and Y are normed spaces and A is unbounded, that is, there exists a sequence of elements Indeed, let A be a linear operator from X into Y with domain Specifically, a complex number λ could be one-to-one but still not bounded below.The stable computation of values of unbounded operators is one of the most important problems in computational mathematics. In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.
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