![]() ![]() (such an extension exists by virtue of Zorn’s lemma). Under hypotheses as above, Equation (4) has a unique solution In the elementary theory of Hilbert and Banach spaces, the linear operatorsthat are considered acting on such spaces or from one such space to another are taken to bebounded, i.e., whenTgoes fromXtoY, it is assumed tosatisfy T xkY CkxkX, for all xX (12.1) this is the same as being continuous. 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 ADJOINT OF UNBOUNDED OPERATORS ON BANACH SPACES 5 Thus, D(B) fu2Y : x7hu Axi Y continuousg and hu Axi Y hBu xi X 8x2D(A) u2D(B): De nition 10. 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 (Some results are new even for Hilbert spaces. (possibly multi-valued) and y is a given element in In this paper, we report on new results related to the exis- tence of an adjoint for operators on separable Banach spacesand discuss few interesting applications. Is a hemi-continuous monotone operator from X into Let X be a real strictly convex reflexive Banach space with the dual The Stable Method of Computing Values of Hemi-Continuous Monotone Operators The approximate values of the operator A atģ. In a similar way as above, the everywhere defined inverse Because of the uniqueness of decomposition (7), x is uniquely determined by z, and so the everywhere defined inverse , we have the uniquely determined decomposition We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely dened linear operators on a separable Banach space can be approximated by bounded. This result is used to extend well known theorems of von Neumann and Lax. Is a closed densely defined linear operator thenĪre complementary orthogonal subspaces of the Hilbert space an adjoint for operators on separable Banach spaces. The following lemma will be used in the proof of Theorem 2.2. To further simplify the presentation, we introduce the functions To establish the convergence of (3), it will be convenient to reformulate (3) asĪre known to be bounded everywhere defined linear operators and 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. 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 Vidav 21 for elements in an arbitrary Banach algebra. In the another case, where A is a monotone operator from a real strictly convex reflexive Banach space X into its dual unbounded self adjoint operator on Hubert space and a concept introduced by I. 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. , 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 If \(\operatorname )g=g\).The stable computation of values of unbounded operators is one of the most important problems in computational mathematics. We now proceed with topological notions for relations. For a relation A ⊆ X 0 × X 1 we will use the abbreviation − A := −1 A (so that the minus sign only acts on the second component). ![]()
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