L. S. RAKOVSHCHIK

SOME PROPOSITIONS ON THE REGULARIZATION OF LINEAR OPERATORS IN BANACH SPACES

(Presented by Academician V. I. Smirnov on 19 V 1961)

1. Let \(E\) and \(E_1\) be Banach spaces and let \(A\) be a closed linear operator from \(E\) into \(E_1\). Denote by \(D(A)\) the domain of definition of the operator; by \(\mathfrak R(A)\), the range of its values; by \(Z_A\), the subspace of its zeros; and by \(Z_A^*\), the subspace of all linear functionals orthogonal to \(\mathfrak R(A)\). Following \((^1)\), we shall call the operator \(A\) a \(\Phi\)-operator (a \(\Phi_+\)-operator) if it is normally solvable* and the subspaces \(Z_A\) and \(Z_A^*\) are finite-dimensional (\(Z_A\) is finite-dimensional, while \(Z_A^*\) is not).

We shall give here several known propositions (see \((^1)\)) that will be needed below. If \(x_1, x_2, \ldots, x_n\) are linearly independent elements of the Banach space \(E\), and \(f_1, f_2, \ldots, f_n\) are functionals biorthogonal to them, then the space \(E\) decomposes into the direct sum \(E=\sigma_1+\sigma_2\), where \(\sigma_1\) is the finite-dimensional subspace with basis \(x_1, x_2, \ldots, x_n\), and \(\sigma_2\) is the subspace of common zeros of the functionals \(f_1, f_2, \ldots, f_n\). To verify the validity of this assertion, it is enough to note that for any \(x\in E\) the element
\[ y=x-\sum_{i=1}^{n} f_i(x)x_i\in\sigma_2 \]
and that, if \(y_1\in\sigma_1\), \(y_2\in\sigma_2\) and \(y_1+y_2=0\), then \(y_1=y_2=0\), since then \(f_i(y_1)=0\) for all \(i\). Suppose now that \(A\) is a \(\Phi_+\)-operator. In this case the subspace \(\overline{D(A)}\) is representable in the form of the direct sum \(\overline{D(A)}=Z_A+\sigma\), where \(\sigma\) is the intersection of the subspace of common zeros of the functionals biorthogonal to a basis of the subspace \(Z_A\) with the subspace \(\overline{D(A)}\). The operator \(A\), considered as an operator from \(\sigma\cap D(A)\) into \(\mathfrak R(A)\), has an inverse operator \(R_0\). Consequently, it is bounded as an operator from \(\mathfrak R(A)\) into \(E\). If \(x_1, x_2, \ldots, x_n\) is a basis in \(Z_A\) and \(f_1, f_2, \ldots, f_n\) are functionals biorthogonal to this basis, then
\[ R_0Ax=x-\sum_{i=1}^{n} f_i(x)x_i. \]

2. As usual, by a regularizer of the operator \(A\) we shall mean a bounded operator \(R\) from \(E_1\) into \(E\) such that \(RA=I+T\), where \(I\) is the identity and \(T\) is a completely continuous operator in \(E\).

Theorem 1. The difference of two regularizers of a \(\Phi\)-operator is a completely continuous operator.

Proof. Let \(f_1, f_2, \ldots, f_n\) be a normalized basis of the subspace \(Z_A^*\), and let \(y_1, y_2, \ldots, y_n\) be elements of the space \(E_1\) biorthogonal to the indicated basis. According to what was said in the preceding paragraph,
\[ E_1=\mathfrak R(A)+\sigma_1, \tag{1} \]

* This is equivalent to the closedness of the linear manifold \(\mathfrak R(A)\).

where \(\sigma_1\) is a subspace with basis \(y_1, y_2,\ldots,y_n\). The decomposition indicated above guarantees the existence of bounded projectors \(P_{\mathfrak R(A)}\) and \(P_{\sigma_1}\) onto the subspaces \(\mathfrak R(A)\) and \(\sigma_1\), respectively.

Let now \(R_1\) and \(R_2\) be two regularizers of the operator \(A\), and let \(R=R_1-R_2\). The operator \(RA\) is defined on all of \(E\) and is completely continuous. Therefore the operator \(R'\), induced by the operator \(R\) on the subspace \(\mathfrak R(A)\), is completely continuous (since \(R'y=RAR_0y\)).

The operator \(R''y\), induced by the operator \(Ry\) on \(\sigma_1\), is also completely continuous, since it is bounded and the subspace \(\sigma_1\) is finite-dimensional. Since
\[ R=R'P_{\mathfrak R(A)}+R''P_{\sigma_1}, \]
the operator \(R\) is also completely continuous.

Corollary 1. The totality of all regularizers of a bounded \(\Phi\)-operator \(A\) is described by the formula
\[ R=R^0+T, \tag{2} \]
where \(R^0\) is an arbitrary fixed regularizer of the operator \(A\), and \(T\) is an arbitrary completely continuous operator from \(E_1\) into \(E\).

Corollary 2. The totality of all regularizers of a closed \(\Phi\)-operator is described by the same formula (2), but the completely continuous operator \(T\) must satisfy the additional condition: the operator \(TA\) is completely continuous.

Corollary 3. The totality of all regularizers of a \(\Phi_+\)-operator is described by the same formula (2), in which \(R^0\) has its former meaning, and \(T\) is a bounded operator from \(E_1\) into \(E\), inducing on \(\mathfrak R(A)\) a completely continuous operator \(T'\) such that the operator \(T'A\) is completely continuous.

1. It was shown by S. G. Mikhlin \((^2)\) that if an operator \(A\) admits a regularization, then it is a \(\Phi_+\)-operator. With the aid of Theorem 1 one can obtain the following result in this direction:

Theorem 2. A bounded operator \(A\) from \(E\) into \(E_1\) admits a regularization (on the left) if and only if: 1) it is a \(\Phi_+\)-operator; 2) there exists a completely continuous operator \(T\) in the space \(\mathfrak R(A)\) such that the operator \(I_{\mathfrak R(A)}+T\) admits an extension to a bounded operator from \(E_1\) into \(\mathfrak R(A)\) (by \(I_\sigma\), here and below, is denoted the identity operator of the space \(\sigma\)). If the operator \(A\) is only closed, then for the existence of a regularizer it is sufficient that conditions 1 and 2 and the condition be fulfilled: the operator \(TA\) is completely continuous.

Corollary. If \(A\) is a \(\Phi_+\)-operator and \(\mathfrak R(A)\) has a direct complement in \(E_1\), then the operator \(A\) admits a regularization.

In particular, the conditions of the corollary are fulfilled if \(A\) is a \(\Phi\)-operator or if the space \(E_1\) is Hilbert (cf. \((^3,^4)\)).

Up to now we have considered only bounded regularizers. If one does not assume boundedness of the regularizer, then the following holds:

Theorem 3. In order that a closed operator admit a regularization, it is necessary and sufficient that it be a \(\Phi_+\)-operator.

1. It was shown by S. G. Mikhlin \((^2)\) that if \(E\) is a \(\Phi\)-operator acting in Hilbert space, then the equation \(A\varphi=f\), in the case of its solvability, is equivalent to a certain Riesz–Schauder equation of the form \(RA\varphi=Rf\). It turns out that this assertion also holds in arbitrary Banach spaces.

Lemma. Let \(A\) and \(R\) be distributive operators and let \(D(R)\supseteq \mathfrak R(A)\). In order that the equations \(A\varphi=f\) and \(RA\varphi=Rf\) be equivalent for elements \(f\) of some set \(M\), it is necessary and sufficient that the equation \(R\psi=0\) have no nontrivial solutions of the form \(\psi=\psi_1-\psi_2\), where \(\psi_1\in\mathfrak R(A)\), and \(\psi_2\in M\).

Proof. Necessity. Let the equations \(A\varphi=f\) and \(RA\varphi=Rf\) be equivalent for all \(f\in M\). Suppose that the equation \(R\psi=0\) has a nontrivial solution \(\psi_0\) of the indicated form, i.e.
\[ \psi_0=A\varphi_0-f_0,\quad \text{where } f_0\in M. \]

Two cases are possible:

1) \(f_0 \notin \mathfrak R(A)\cap M\). In this case the equation \(A\varphi=f_0\) is unsolvable, whereas the equation \(RA\varphi=Rf_0\) has the solution \(\varphi_0\).

2) \(f_0\in \mathfrak R(A)\cap M\). In this case \(\psi_0=A\varphi_0-A\varphi_1\ne0\), so that \(A\varphi_0\ne A\varphi_1\). The equation \(RA\varphi=RA\varphi_1\) has the solution \(\varphi_0\), which is not a solution of the equation \(A\varphi=A\varphi_1\).

In both cases we arrive at a contradiction, which proves the necessity of the condition of the lemma.

Sufficiency. Suppose that the conditions of the lemma are fulfilled and \(f\in M\). Consider the equation \(RA\varphi=Rf\). It is equivalent to the equation \(R(A\varphi-f)=0\). If this equation has a solution \(\varphi_0\), then \(A\varphi_0=f\), since the equation \(R\psi=0\) has no nontrivial solutions of the form \(A\varphi_0-f\in M\).

Corollary. In order that the equations \(A\varphi=f\) and \(RA\varphi=Rf\) be equivalent, in the case of solvability of the first of them it is necessary and sufficient that the equation \(R\psi=0\) have no nontrivial solutions in the range of the operator \(A\). In order that the indicated equations be equivalent for all \(f\), it is necessary and sufficient that the equation \(R\psi=0\) have no nontrivial solutions.

Theorem 4. If \(A\) is a \(\Phi_+\)-operator, then the equation \(A\varphi=f\), in the case of solvability, is equivalent to a certain Riesz–Schauder equation.

Proof. Let \(R_0\) be the operator constructed in § 1. This operator is initially defined only on \(\mathfrak R(A)\). In the usual way we extend the operator \(R_0\) to a distributive operator \(R\) (generally speaking, unclosed) defined on the whole space \(E_1\). By construction, the extended operator \(R\) has no zeros in \(\mathfrak R(A)\). Therefore the equations \(A\varphi=f\) and \(RA\varphi=Rf\) are equivalent for \(f\in\mathfrak R(A)\). It remains to observe that

\[ RA\varphi=R_0A\varphi=\varphi \sum_{i=1}^{\dim Z_A} f_i(\varphi)x_i, \]

i.e., that \(RA\) is a Riesz–Schauder operator.

Corollary 1. If the space \(E_1\) decomposes into the direct sum of its subspaces \(\mathfrak R(A)\) and \(\sigma_1\), then the equation \(A\varphi=f\), in the case of its solvability, can be reduced to an equivalent Riesz–Schauder equation by the application of a bounded regularizer.

Indeed, putting

\[ R\varphi= \begin{cases} R_0\varphi, & \varphi\in\mathfrak R(A),\\ 0, & \varphi\in\sigma_1, \end{cases} \]

we obtain a bounded extension of the operator \(R_0\) to the whole space \(E_1\). The decomposition indicated in this corollary occurs, for example, if \(E_1\) is a Hilbert space.

Corollary 2. If \(A\) is a \(\Phi\)-operator, then the equation \(A\varphi=f\), in the case of its solvability, reduces to an equivalent Riesz–Schauder equation by the application of a bounded regularizer.

From Corollary 2, in particular, follows the theorem of S. G. Mikhlin mentioned at the beginning of this section.

The author expresses his gratitude to Prof. S. G. Mikhlin for advice and discussion.

Petrozavodsk State
University

Received
4 V 1961

CITED LITERATURE

1. I. Ts. Gokhberg, M. G. Krein, UMN, 12, no. 2, 43 (1957).
2. S. G. Mikhlin, UMN, 3, no. 3, 30 (1948).
3. F. Atkinson, Matem. sborn., 26, no. 1, 3 (1951).
4. I. Ts. Gokhberg, DAN, 76, no. 4, 477 (1951).