MATHEMATICS

I. Ts. Gohberg, M. G. Krein

On the Problem of Factorization of Operators in Hilbert Space

(Presented by Academician V. I. Smirnov on 1 VI 1962)

1. As is known \((^1)\), Gauss’s method for inverting a nonsingular matrix \(A=\|a_{jk}\|_1^n\) shows that, under certain conditions, the inverse matrix \(A^{-1}\) can be represented, moreover uniquely, in the form of a product of three factors
\(A^{-1}=S_+DS_-\), where the middle one is a diagonal matrix, and the outer ones \(S_\pm\) are right and left triangular matrices with diagonal elements equal to one. Such a representation is possible if and only if all principal minors \(D_r=|a_{jk}|_1^r\ne0\) \((r=1,2,\ldots,n)\). In operator form, the indicated factors can be obtained by the formulas

\[ S_+=\sum_{j=1}^{n}\frac{D_j}{D_{j-1}}(P_jAP_j)^{(-1)}\Delta P_j,\qquad S_-=\sum_{j=1}^{n}\frac{D_j}{D_{j-1}}\Delta P_j(P_jAP_j)^{(-1)}, \]

\[ D=\sum_{j=1}^{n}\frac{D_{j-1}}{D_j}\Delta P_j. \]

Let us explain the notation. With each matrix of order \(n\) there is associated an operator acting in the \(n\)-dimensional space \(E_n\) of complex vectors \(\xi=\{\xi_j\}_1^n\) and denoted by the same letter as the matrix. By \(P_j\) \((j=1,2,\ldots,n)\) is denoted the projection operator in \(E_n\) which preserves the first \(j\) coordinates of the vector \(\xi\) and annihilates all the others; \(\Delta P_j=P_j-P_{j-1}\) \((j=1,2,\ldots,n;\ P_0=0)\). Finally, \((P_jAP_j)^{(-1)}\) \((j=1,2,\ldots,n)\) denotes the operator acting in the subspace \(P_jE_n\) and there inverse to the operator \(P_jAP_j\) (it corresponds to the matrix inverse to the truncated matrix \(\|a_{lk}\|_1^j\)).

In the case when \(A\) is a Hermitian matrix, necessarily \(S_-=S_+^*\), and the representation \(A^{-1}=S_+DS_+^*\) itself is easily obtained from the expansion of the form \((A\xi,\xi)\) into a sum of squares by the Lagrange–Jacobi method.

There is also known a continuous analogue of the indicated matrix factorization for a Fredholm integral operator with kernel \(\delta(x-s)-K(x,s)\) \((a\le x,s\le b)\), where \(\delta(x)\) is the Dirac function and \(K(x,s)\), for example, is a continuous kernel. The role of the condition \(D_j\ne0\) \((j=1,2,\ldots,n)\) is now played by the condition that, for every \(c\) \((a<c\le b)\), the integral equation

\[ \varphi(x)-\int_a^c K(x,s)\varphi(s)\,ds=f(x)\quad (a\le x\le c) \tag{1} \]

has the resolvent \(\Gamma_c(x,s)\). We form the Volterra kernels \(V_\pm(x,s)\), putting
\(V_-(x,s)=\Gamma_x(x,s)\) \((a\le s\le x\le b)\) and \(V_-(x,s)=0\) \((a\le x<s\le b)\), and similarly
\(V_+(x,s)=\Gamma_s(x,s)\) \((a\le x\le s\le b)\) and \(V_+(x,s)=0\) \((a\le s<x\le b)\). Then

\[ \delta(x-s)+\Gamma_b(x,s)=\bigl(\delta(x-s)+V_+(x,s)\bigr)*\bigl(\delta(x-s)+V_-(x,s)\bigr), \tag{2} \]

where the symbol \(*\) denotes the operation of composition of kernels defined in the square \(a \leqslant x, s \leqslant b\). This factorization is a continuous analogue of the matrix factorization with three factors considered above; the diagonal factor in representation (2) is, naturally, absent, since the role of \(D_i\) is now played by the Fredholm determinant of equation (1), and therefore the ratios \(D_{i-1}/D_i\) correspond to unity. Equality (2) has been used for various purposes in papers \((2\text{–}4)\). Below we shall give continuous analogues and generalizations of the indicated factorizations for various classes of operators in Hilbert space.

1. Let \(\mathfrak R\) denote the ring of all bounded linear operators acting in a Hilbert space \(\mathfrak H\). Let \(\mathfrak P=\{P\}\) be an arbitrary maximal continuous chain of orthogonal projectors \((^5)\). The chain \(\mathfrak P\) is called a proper chain of an operator \(A\in\mathfrak R\) if all subspaces \(P\mathfrak H\) \((P\in\mathfrak P)\) are invariant subspaces of the operator \(A\), i.e. \(PAP=AP\) \((P\in\mathfrak P)\). By \(\mathfrak P^\perp\) we denote the chain consisting of all projectors \(I-P\) \((P\in\mathfrak P)\). It is obvious that the sets \(\mathfrak R^+(\mathfrak P)\) and \(\mathfrak R^-(\mathfrak P)=\mathfrak R^+(\mathfrak P^\perp)\) of all operators possessing, respectively, the chains \(\mathfrak P\) and \(\mathfrak P^\perp\), are closed subrings of the ring \(\mathfrak R\).

In accordance with \((^5)\), by \(\mathfrak S_\infty\) we shall denote the ideal of all completely continuous operators from \(\mathfrak R\), and by \(\mathfrak S_\infty^+(\mathfrak P)\) and \(\mathfrak S_\infty^-(\mathfrak P)\) the ideals of the subrings \(\mathfrak R^+(\mathfrak P)\) and \(\mathfrak R^-(\mathfrak P)\) that are their intersection with \(\mathfrak S_\infty\).

By a factorization of an operator \(A\in\mathfrak R\) with respect to a chain \(\mathfrak P\) we shall mean a representation of the operator \(A\) in the form of a product: \(A=A_+A_-\), where \(A_\pm\in\mathfrak R^\pm(\mathfrak P)\). If a factorization for a given \(A\) \((\in\mathfrak R)\) is possible, then it is not unique. Indeed, from a given factorization one can obtain an infinite set of others by putting, for example, \(A'_+=A_+B\) and \(A'_-=B^{-1}A\), where \(B\) is an arbitrary invertible operator commuting with all projectors \(P\in\mathfrak P\). However, if a factorization is possible which in what follows we shall call special:

\[ A=(I+X_+)(I+X_-)\qquad \bigl(X_\pm\in\mathfrak S_\infty^\pm(\mathfrak P)\bigr), \tag{3} \]

then it will already be unique. Indeed, in this case \(X_\pm\) will be Volterra operators \((^5,^6)\), and consequently the operators \(I+X_\pm\) (and together with them the operator \(A\)) will be invertible. If, moreover, one takes into account that the intersection of \(\mathfrak S_\infty^+(\mathfrak P)\) and \(\mathfrak S_\infty^-(\mathfrak P)\) consists only of the zero operator, then from this it is already easy to conclude the uniqueness of the factorization (3). From this uniqueness it follows that if a self-adjoint operator \(A\) \((A=A^*)\) admits the special factorization (3), then in it \(X_-=X_+^*\). Therefore such an operator \(A\) is always positive. We also note that, when seeking a formula for the operators \(X_\pm\), without loss of generality we may immediately assume that the operator \(A\) has the form \(A=(I-T)^{-1}\), where \(T\in\mathfrak S_\infty\).

1. Let an arbitrary operator-function \(F(P)\) be given, defined on the chain \(\mathfrak P\) and taking its values as operators from \(\mathfrak R\). With its aid one can form a function \(S(\mathfrak z)\) of partitions (multivalued), putting for each partition \(\mathfrak z=\{0=P_0<P_1<\cdots<P_n=I;\; P_j\in\mathfrak P\}\) of the chain \(\mathfrak P\):

\[ S(\mathfrak z)=\sum_{j=1}^{n} F(Q_j)(P_j-P_{j-1}) \qquad (P_{j-1}\leqslant Q_j\leqslant P_j;\; Q_j\in\mathfrak P). \tag{4} \]

If \(S(\mathfrak z)\) has as its limit (in the sense of S. Shatunovskii) in the norm of operators some operator \(Y\) \((\in\mathfrak R)\), i.e. if for every \(\varepsilon>0\) there exists a partition \(\mathfrak z_\varepsilon\) of the chain \(\mathfrak P\) such that for all partitions \(\mathfrak z\) of this chain containing \(\mathfrak z_\varepsilon\), every value of the function \(S(\mathfrak z)\) satisfies the condition

\(\|S(\delta)-Y\|<\varepsilon\), then we shall write

\[ Y=\int_{\mathfrak P} F(P)\,dP \tag{5} \]

and say that the integral appearing on the right-hand side of (5) converges \((^{5,6})\).

Along with the integral (5) we shall also need the integral

\[ Y=\int_{\mathfrak P}^{+} F(P)\,dP, \]

whose definition differs from the definition of the integral (5) only in that, in the equality (4) defining the function \(S(\delta)\), the projector \(Q_j\) is chosen in a completely definite way, namely \(Q_j=P_j\) \((j=1,2,\ldots,n)\).

Theorem 1. In order that the operator \(A=(I-T)^{-1}\) \((A\in\mathfrak R,\ T\in\mathfrak S_\infty)\) admit a special factorization with respect to a continuous maximal chain \(\mathfrak P\), it is necessary and sufficient that: a) all operators \(I-PTP\) \((P\in\mathfrak P)\) be invertible, and b) at least one of the integrals

\[ X_+=\int_{\mathfrak P}(I-PTP)^{-1}PT\,dP,\qquad X_-=\int_{\mathfrak P} dP\,TP(I-PTP)^{-1} \tag{6} \]

converge.

If at least one of these integrals converges, then the other also converges, and their values give the special factorization (3).

Let us also note that in the formulation of Theorem 1 the integrals (6) may be replaced by the following:

\[ X_+=\int_{\mathfrak P}^{+}\bigl[(I-PTP)^{-1}-I\bigr]\,dP,\qquad X_-=\int_{\mathfrak P}^{+} dP\,\bigl[(I-PTP)^{-1}-I\bigr]. \]

1. A normed ideal \((^{7})\) of the ring \(\mathfrak R\) is any ideal \(\mathfrak S\) of the ring \(\mathfrak R\) in which its own norm \(|X|_{\mathfrak S}\) \((X\in\mathfrak S)\) is defined, turning it into a Banach space and such that

\[ |XAY|_{\mathfrak S}\leq \|X\|\,|A|_{\mathfrak S}\,\|Y\| \qquad (X,Y\in\mathfrak R,\ A\in\mathfrak S). \]

Theorem 2. Let \(\mathfrak P\) be some continuous maximal chain, and let \(\mathfrak S_{\mathrm I}\) and \(\mathfrak S_{\mathrm{II}}\) be two normed ideals of the ring \(\mathfrak R\).

If the following conditions are satisfied: 1) \(\mathfrak S_{\mathrm I}\subset \mathfrak S_{\mathrm{II}}\); 2) the finite-dimensional operators form a dense set in \(\mathfrak S_{\mathrm I}\); and 3) for every \(T\in\mathfrak S_{\mathrm I}\) the integral

\[ S=\int_{\mathfrak P} PT\,dP \]

converges and belongs to \(\mathfrak S_{\mathrm{II}}\), then for every \(T\in\mathfrak S_{\mathrm I}\) satisfying the necessary condition a) (from Theorem 1), the integrals (6) converge, the operators \(X_\pm\in\mathfrak S_{\mathrm{II}}\), and, consequently, the operator \(A=(I-T)^{-1}\) admits the special factorization (3) with operators \(X_\pm\in\mathfrak S_{\mathrm{II}}\).

It is not difficult to see that if an operator \(T\in\mathfrak S_{\mathrm I}\) satisfies the necessary condition a), then all operators in some neighborhood of it in \(\mathfrak S_{\mathrm I}\) will also satisfy this condition, and the integrals (6) will map this neighborhood continuously into \(\mathfrak S_{\mathrm{II}}\).

By virtue of the results \((^{5,8-11})\), it follows from Theorem 2 that as \(\mathfrak S_{\mathrm I}\) one may, for example, take the normed ideals \(\mathfrak S_1,\ \mathfrak S_p\) \((1<p<\infty),\ \mathfrak S_\omega\), and then as \(\mathfrak S_{\mathrm{II}}\), respectively, the ideals \(\mathfrak S_\Omega,\ \mathfrak S_p,\ \mathfrak S_\infty\).

From Theorem 2 and what has been said we obtain:

Corollary. If the operator \(I-H\) \((H\in\mathfrak S_\omega)\) is positive \(((Hf,f)<(f,f),\ f\ne0)\), then it admits a special factorization with respect to any continuous maximal chain \(\mathfrak P\).

Apparently, this proposition is “sharp” in the sense that if \(H\) does not belong to \(\mathfrak S_\omega\), then there will be found a chain \(\mathfrak P\) with respect to which the special factorization of the operator \(I-H\) proves impossible.

1. In the proof of Theorem 2 an essential role is played by the following general

Lemma. Let \(R\) be an arbitrary normed ring with identity \(e\) \((|e|_R=1)\), containing two closed subrings \(R_{\pm}\) which intersect only in zero. Suppose, moreover, that in \(R\) there exists a right (left) ideal \(J\) with the following properties: a) \(J\) is provided with its own norm \(|c|_J\) \((c\in J)\), making \(J\) a Banach space; b) there exists a constant \(k(>0)\) such that
\[ |cx|_J \le k|c|_J|x|_R \quad \bigl(|xc|_J \le k|c|_J|x|_R\bigr) \quad (x\in R,\ c\in J); \]
c) every element \(c\in J\) can be represented uniquely in the form \(c=c_+ + c_-\), where \(c_{\pm}\in R_{\pm}\), and, moreover,*
\[ \sup_{c\in J}\bigl(|c_+|_R/|c|_J\bigr)<\infty . \]

Then there exists a number \(\delta\) \((0<\delta<1)\) such that, for every \(a\in J\) with norm \(|a|_J<\delta\), the element \((e-a)^{-1}\) can be represented in the form
\[ (e-a)^{-1}=(e+b_+)(e+b_-)\qquad (b_{\pm}\in R_{\pm}), \tag{7} \]
and if \(J\) is a right ideal, then
\[ b_+ = a_+ + (aa_+)_+ + \bigl(a(aa_+)_+\bigr)_+ + \cdots, \tag{8} \]
while if \(J\) is a left ideal, then
\[ b_- = a_- + (a_-a)_- + \bigl((a_-a)_-a\bigr)_- + \cdots . \tag{9} \]
The convergence of the series (8) and (9) is understood in the sense of convergence in the norm of the ring \(R\).

In proving Theorem 2 the lemma is used for the case when \(J\) is a two-sided ideal in \(R\) and when, consequently, the decompositions (8) and (9) hold simultaneously. Propositions akin to the lemma in content and proof are found in papers \((^{12,13})\).

In the proof it turns out that the number \(\delta\) can be chosen so small that the elements \(e+b_{\pm}\) are invertible and \((e+b_{\pm})^{-1}-e\in R_{\pm}\).

Let the set \(D\) consist of all elements \(a\in J\) for which the element \(e-a\) is invertible and \((e-a)^{-1}\) admits a factorization (7) with invertible factors \(e+b_{\pm}\), for which \((e+b_{\pm})^{-1}-e\in R_{\pm}\). Such a factorization for \(a\in D\) will be unique. From the lemma one can derive that, under the conditions specified in it, \(D\) is an open part of \(J\) and is mapped continuously into \(R_{\pm}\) by the correspondence operators \(a\to b_+\) and \(a\to b_-\).

1. The results presented are generalized to the case of factorization of operators with three factors relative to an arbitrary discontinuous maximal chain \(\mathfrak P\). They make it possible to obtain triangular representations for operators “close” to unitary ones, which will be set forth elsewhere.

The factorizations used in the theory of scalar and vector Wiener—Hopf integral equations and their discrete analogues \((^{14,15})\) are also examples of factorization in the sense of the general definition given in § 1, although the operators considered in these theories are bounded operators of a different type than in Theorems 1 and 2.

Odessa Civil Engineering Institute

Institute of Physics and Mathematics
Academy of Sciences of the Moldavian SSR

Received
29 V 1962

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\[ \text{* This condition will certainly be satisfied if }\sup_{c\in J}\bigl(|c|_R/|c|_J\bigr)<\infty . \]