UDC 517.948.35

MATHEMATICS

A. B. BAKUSHINSKII

REGULARIZATION ALGORITHMS FOR LINEAR EQUATIONS WITH UNBOUNDED OPERATORS

(Presented by Academician A. N. Tikhonov on 4 III 1968)

Consider the operator equation:

\[ Ax=f, \tag{1} \]

where \(A\) is a linear closed operator with an everywhere dense domain of definition in some Hilbert space \(H_1\), acting into the Hilbert space \(H_2\), having no bounded inverse, and \(f \in A(D_A)\). The spaces \(H_1\) and \(H_2\) are assumed to be complete and real.

Interest in the approximate solution of such equations has recently increased thanks to the works of A. N. Tikhonov (for example \((^1,^2)\)). In these works the problem of solving (1) was reduced to the problem of constructing a regularizing algorithm. This reduction proved convenient, in particular, for the numerical solution of equation (1).

In what follows, by a regularizing algorithm (r.a.) for equation (1) we shall mean a family of linear bounded operators \(\{R\}\), acting from \(H_2\) to \(H_1\), such that for any \(f \in A(D_A)\) one can indicate \(x \in A^{-1}f\) such that for any \(\varepsilon > 0\) there exists \(\delta=\delta(f,\varepsilon,A)\) and an operator \(R\) from our family such that \(\|R\tilde f-x\|\le \varepsilon\), provided only that \(\|\tilde f-f\|\le \delta\).

In \((^3)\) an abstract analogue of the r.a. considered in \((^2)\) was extended to operator equations (1), under the assumption that \(H_1=H_2\) and \(f\in D(A^*)\). The aim of the present work is to generalize the scheme for obtaining an r.a. proposed in our work \((^4)\) for bounded operators \(A\), to equations with unbounded operators. The r.a. studied in \((^{1-3})\) enters into this scheme as a special case.

First of all, let us note that, by the theorem of J. von Neumann \((^5)\), there exists a self-adjoint operator \(A^*A\) with dense domain of definition \(D(A^*A)\), acting from \(H_1\) to \(H_1\). Denote by \(\{E_\lambda\}\) its resolution of the identity.

Let the real function \(\psi(\lambda,\alpha)\) possess the following properties: it is defined for \(\lambda \in S(A^*A)\) and \(\alpha>0\) as follows:

\[ \psi(\lambda,\alpha)= \begin{cases} \varphi(\lambda,\alpha)/\lambda, & \lambda\ne 0,\\ K, \quad |K|<\infty, & \lambda=0. \end{cases} \tag{2} \]

\(\varphi(\lambda,\alpha)\) is bounded with respect to \(\lambda\) and \(\alpha\), measurable in \(\lambda\) for each \(\alpha\) with respect to \(\{E_\lambda\}\); moreover,

a) \(\displaystyle \lim_{\alpha\to 0}\varphi(\lambda,\alpha)=1,\quad \varphi(0,\alpha)=0;\)

b) \(\displaystyle \sup_{\substack{\lambda\in S(A^*A)\\ \lambda\ne 0}} |\varphi(\lambda,\alpha)|/\sqrt{\lambda}=K_\alpha<\infty \quad (\alpha>0),\)

where in a) the convergence to the limit is uniform on the set \(S(A^*A)\cap(c,\infty)\), where \(c\) is any positive number.

Denote

\[ \Omega_k=\{\lambda:\ k-1\le \psi(\lambda,\alpha)<k\},\quad k=\ldots,-2,-1,0,1,\ldots \]

By our assumptions, every set \(\Omega_k\) is measurable with respect to \(\{E_\lambda\}\).

Let \(E(\Omega_k)\) be its spectral measure. Define the operator \(B_{k\alpha}\) by the formula

\[ B_{k\alpha}=\overline{\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda E(\Omega_k)A^*}. \tag{3} \]

(The bar denotes the closure of an operator. The possibility of taking the closure will be shown below.)

The operator \(B_{k\alpha}\) is bounded and \(\|B_{k\alpha}\|\le K_\alpha\). Indeed, the operator
\[ \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda E(\Omega_k)A^* \]
is defined on an everywhere dense set in \(H_2\) and has an adjoint given by the formula

\[ AE(\Omega_k)\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda = A\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda E(\Omega_k). \tag{4} \]

Moreover, on a dense set in \(H_1\) the equality

\[ \left( AE(\Omega_k)\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda x,\, AE(\Omega_k)\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda x \right) = \left( \int_{\Omega_k}\lambda\psi^2(\lambda,\alpha)\,dE_\lambda x,\,x \right) \tag{5} \]

holds.

Indeed, if \(x\in \overline{H}_{1k}=E(\Omega_k)H_1\), then equality (5) is valid (both sides are equal to \(0\)). If \(x\in H_{1k}\), then the operator
\[ \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda \]
is invertible in the subspace \(H_{1k}\) \((k\ne 0,1)\). In the case \(k=0\) (or \(1\)) the space \(H_{1k}\) itself can be decomposed into a direct sum of its invariant subspaces with respect to
\[ \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda, \]
in each of which the operator
\[ \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda \]
is invertible. Consequently, since \(D(A^*A)\) is dense in \(H_{1k}\), there exists a dense set in \(H_{1k}\) (respectively, in each of the subspaces forming \(H_{1k}\)) on which the operator

\[ \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda E(\Omega_k)A^*AE(\Omega_k) \int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda = \int_{\Omega_k}\lambda\psi^2(\lambda,\alpha)\,dE_\lambda \]

is defined.

From (5) and the closedness of the operator (4) it follows that (4) is bounded; moreover, by condition b) its norm is \(\le K_\alpha\). The adjoint of (4) is equal to \(B_{k\alpha}\). Therefore, \(\|B_{k\alpha}\|\le K_\alpha\). Finally, note that the ranges of the operators \(B_{k\alpha}\) for different \(k\) are orthogonal. Define the operator

\[ B_\alpha=\sum_{k=-\infty}^{\infty}\int_{\Omega_k}\psi(\lambda,\alpha)\,dE_\lambda E(\Omega_k)A^*. \tag{6} \]

By what was said above, the operator (6) is bounded. It is easy to show that the exact equality \(\|B_\alpha\|=K_\alpha\) holds.

We shall now prove that the family \(\{B_\alpha\}\) forms a r.a. for equation (1). Let \(f\in A(D_A)\). Take such a solution \(\hat{x}\) of (1) for which the condition \(\hat{x}\perp \ker A^*A\) is satisfied. This means that the function \((E_\lambda \hat{x},\hat{x})\) is continuous at \(\lambda=0\). Note that
\[ B_\alpha A\hat{x}=\int_0^\infty \varphi(\lambda,\alpha)\,dE_\lambda \hat{x}. \]
Consider \(B_\alpha\tilde{f}-\hat{x}\). We have

\[ \|B_\alpha\tilde{f}-\hat{x}\| \le \|B_\alpha A\hat{x}-\hat{x}\|+K_\alpha\|\tilde{f}-f\| = \]

\[ = \left( \int_0^\infty(\varphi(\lambda,\alpha)-1)^2\,d(E_\lambda\hat{x},\hat{x}) \right)^{1/2} + K_\alpha\|\tilde{f}-f\|. \tag{7} \]

By virtue of condition a) and the continuity of \((E_{\lambda}\hat{x}, \hat{x})\) at \(\lambda = 0\), for any \(\varepsilon > 0\) one can choose such an \(\alpha\) (and consequently also \(B_{\alpha}\)) and such a \(\delta\) that \(\|B_{\alpha}\tilde{f} - \hat{x}\| \leqslant \varepsilon\), provided only that \(\|\tilde{f} - f\| \leqslant \delta\). All the specific functions \(\psi(\lambda,\alpha)\) (with the exception of \(\psi(\lambda,\alpha) = [1 - (1-\mu\lambda)^{1/\alpha}]/\lambda\) for \(\lambda \ne 0\); \(\psi(\lambda,\alpha) = \mu/\lambda\) for \(\lambda = 0\)), considered in \((^4)\), are suitable for constructing r.a. in our case. In particular, for (1) one can also construct an “optimal” r.a. \((^4)\). In the case when \(H_1 = H_2\) and \(A\) is self-adjoint, the scheme for obtaining r.a. can be substantially simplified. The results of \((^4)\) carry over to equation (1) in this case without any changes.

An analogous scheme for obtaining r.a. in this special case was also proposed in \((^6)\).

The r.a. obtained above may be applied, for example, to constructing an approximate solution of unstable boundary-value problems for differential equations under the condition that such a problem is solvable, and in other analogous cases.

Moscow State University
named after M. V. Lomonosov

Received
1 III 1968

REFERENCES

\(^1\) A. N. Tikhonov, DAN, 151, No. 3, 501 (1963).
\(^2\) A. N. Tikhonov, DAN, 153, No. 1, 49 (1963).
\(^3\) V. P. Maslov, Perturbation Theory and Asymptotic Methods, Moscow, 1965.
\(^4\) A. B. Bakushinskii, Zh. Vychisl. Mat. i Mat. Fiz., 7, No. 3, 672 (1967).
\(^5\) M. A. Naimark, Normed Rings, Moscow, 1966.
\(^6\) Yu. T. Antokhin, Differential Equations, 3, No. 7, 1135 (1967).