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MATHEMATICS
I. Ts. Gokhberg
CRITERIA FOR ONE-SIDED INVERTIBILITY OF ELEMENTS OF NORMED RINGS AND THEIR APPLICATIONS
(Presented by Academician P. S. Aleksandrov, 20 III 1962)
In the present communication a new general ring-theoretic approach is proposed for the solution and investigation of certain classes of equations, including one-dimensional singular integral equations, systems of equations of the form
\[ \sum_{j=1}^{\infty} \alpha_{k-j}\xi_j=\eta_k \qquad (k=1,2,\ldots) \tag{1} \]
and those close to them. It turns out that the main propositions, for example concerning the system (1), established by M. G. Krein \((^1)\), are in essence criteria for one-sided invertibility of certain classes of elements of a normed ring. This approach has made it possible to obtain new general criteria for one-sided invertibility of elements, from which, in particular, generalizations of the theorems of M. G. Krein \((^1)\) on the system (1) and generalizations of the principal theorems on other classes of equations, in a certain sense close to the system of equations (1), follow.
§ 1.1. In what follows \(R\) denotes an arbitrary normed ring with unit \(e\).
Let \(v \in R\) be an element having an inverse \(v^{(-1)}\) \((\in R)\) strictly on the left: \(v^{(-1)}v=e,\ vv^{(-1)}\ne e\). In what follows we shall assume that \(|v|=|v^{(-1)}|=1\)*. Introduce the following notation: \(K_{(v)}\) is the linear hull of all elements \(v^{(j)}\) \((j=0,\pm1,\ldots)\), where \(v^{(j)}=v^j\) \((j=0,1,\ldots)\) and \(v^{(j)}=(v^{(-1)})^{-j}\) \((j=-1,-2,\ldots)\); \(K^+_{(v)}\), \(K^-_{(v)}\) are respectively the linear hulls of \(v^{(j)}\) \((j=0,-1,\ldots)\) and \(v^{(j)}\) \((j=0,-1,\ldots)\); \(R_{(v)}\), \(R^+_{(v)}\), and \(R^-_{(v)}\) are the closures (in the norm of the ring \(R\)) respectively of \(K_{(v)}\), \(K^+_{(v)}\), and \(K^-_{(v)}\).
It is obvious that \(R^{\pm}_{(v)}\) are commutative normed subrings of \(R\), while \(R_{(v)}\) is a subspace of \(R\). To each element \(r=\sum \alpha_j v^{(j)}\in K_{(v)}\) we associate the function \(r(\zeta)=\sum \alpha_j\zeta^j\) \((|\zeta|=1)\). On the linear space \(K_{(v)}\) define a new multiplication by putting \(r_1\circ r_2=r\), where the element \(r\in K_{(v)}\) is determined by the equality \(r(\zeta)=r_1(\zeta)r_2(\zeta)\). It is easily proved that for any pair of elements \(r_1,r_2\in K_{(v)}\): \(|r_1\circ r_2|\le |r_1|\,|r_2|\). Hence it follows that the new multiplication operation can be extended by continuity to all pairs of elements of \(R_{(v)}\). With this definition of multiplication, \(R_{(v)}\) becomes a normed commutative ring with two mutually inverse generators \(v\) and \(v^{(-1)}\). The bicompactum of maximal ideals of the ring \(R_{(v)}\) is homeomorphic to the entire unit circle. By \(a(\zeta)\) \((|\zeta|=1)\) we denote the function on the bicompactum of maximal ideals of the ring \(R_{(v)}\) corresponding to the element \(a\in R_{(v)}\).
Theorem 1. In order that an element \(a\in R_{(v)}\) be invertible in \(R\) at least on one side, it is necessary and sufficient that the condition
\[ a(\zeta)\ne 0 \qquad (|\zeta|=1). \tag{2} \]
be satisfied.
* The condition \(|v|=|v^{(-1)}|=1\) may be replaced by the condition \(\sup |v^{(j)}|<\infty\).
If condition (2) is satisfied, then for \(\varkappa(a)>0\), where
\[ \varkappa(a)=\frac{1}{2\pi}\,[\arg a(e^{i\varphi})]_{\varphi=0}^{\varphi=2\pi}, \]
the element \(a\) is invertible in \(R\) strictly from the left; for \(\varkappa(a)<0\) it is invertible in \(R\) strictly from the right; and for \(\varkappa(a)=0\) the element \(a\) has in \(R\) a two-sided inverse.
In the proof of this theorem an effective construction is given of the element \(a^{(-1)}\), inverse from the corresponding side to the element \(a\). The formula expressing the relation between the elements \(a^{(-1)}\) and \(a\) is considerably simplified if every element \(y\in R_{(v)}\) can be represented in the form
\[ y=y_+ + y_- \quad (y_+\in R^+_{(v)},\; y_-\in R^-_{(v)}). \tag{3} \]
Namely, the following holds.
Theorem 2. In order that every element \(a\in R_{(v)}\) satisfying condition (2) admit a factorization, i.e. admit a representation in the form
\[ a=a_-\,v^{(\varkappa)}\,a_+ \quad (\varkappa=\varkappa(a)), \]
where \(a_\pm\) are invertible elements respectively of the rings \(R^\pm_{(v)}\), it is necessary and sufficient that every element \(y\in R_{(v)}\) be representable in the form (3).
If condition (3) is fulfilled, then the element \(a^{(-1)}\) is given by the formula
\[ a^{(-1)}=a_+^{-1}\,v^{(-\varkappa)}\,a_-^{-1}. \]
This theorem is a generalization of M. G. Krein’s theorem \((^1)\) on the factorization of functions on the unit circle; its proof is based on a theorem of G. E. Shilov \((^2),\S 13\).
§ 1,2. Let \(\mathfrak B\) be a Banach space, \(\mathfrak R\) the ring of all linear bounded operators acting in \(\mathfrak B\), and let \(V\) be an isometric operator from \(\mathfrak R\), possessing a left inverse \(V^{(-1)}\) such that \(\lvert V^{(-1)}\rvert=1\).* It is clear that Theorems 1 and 2 are applicable to the elements of the ring \(\mathfrak R_{(V)}\). We note, incidentally, that if \(\mathfrak B=\mathfrak h\) is a Hilbert space, then for every \(A\in R_{(V)}\)
\[ \lvert A\rvert=\max_{|\zeta|=1}\lvert A(\zeta)\rvert, \]
and, consequently, in this case the set of all functions \(A(\zeta)\) \((|\zeta|=1;\, A\in\mathfrak R_{(V)})\) consists of all continuous functions on the unit circle.
For the ring \(\mathfrak R_{(V)}\) in the general case of a Banach space \(\mathfrak B\), Theorems 1 and 2 admit the following refinement:
Theorem 3. In order that an operator \(A\in\mathfrak R_{(V)}\) be normally solvable and that at least one of the equations \(Ax=0\), \(A^*f=0\) \((x\in\mathfrak B;\; f\in\mathfrak B^*)\) have a finite number of linearly independent solutions, it is necessary that the condition
\[ A(\zeta)\ne 0 \quad (|\zeta|=1) \tag{4} \]
be satisfied.
If condition (4) is satisfied and \(\varkappa(A)<0\), then the subspace of all solutions \(\mathfrak Z\) of the equation \(Ax=0\) has dimension \(\alpha|\varkappa(A)|\), where \(\alpha\) \((\leq \infty)\) is the dimension of the subspace of all solutions of the equation \(Vx=0\). If \(\alpha<\infty\), then there exists an \(\alpha\)-dimensional subspace \(\Omega\) such that
\[ \mathfrak Z=\Omega \dotplus V\Omega \dotplus \cdots \dotplus V^{m-1}\Omega \quad (m=|\varkappa(A)|). \]
The results listed above, with some additions, were obtained earlier by M. G. Krein \((^1)\) for the following case: \(\mathfrak B\) is one of the spaces of sequences \(\xi=\{\xi_j\}_1^\infty\) of complex numbers \((l_p)\) \((1\leq p<\infty)\), \((m)\),
* In order that an isometric operator \(V\) have a left inverse \(V^{(-1)}\) such that \(\lvert V^{(-1)}\rvert=1\), it is necessary and sufficient that there exist in \(\mathfrak R\) a projector \(P\) with norm equal to one for which \(P\mathfrak B=V\mathfrak B\).
\((c), (c_0);\) \(V\) is the shift operator: \(V\{\xi_1,\xi_2,\ldots\}=\{0,\xi_1,\xi_2,\ldots\}\), and \(V^{(-1)}\) is defined by the equality \(V\{\xi_j\}_1^\infty=\{\xi_{j+1}\}_1^\infty\). In addition, in \((1)\) it was also assumed that the operator \(A\in \mathfrak{R}_{(V)}\) has the form*
\[ A=\sum_{j=-\infty}^{\infty}\alpha_j V^{(j)} \qquad \left(\sum_j |\alpha_j|<\infty\right). \tag{5} \]
With the aid of Theorems 1–3 one can show that in this case the entire theory constructed by M. G. Krein \((^1)\) remains valid if the range space \(\mathfrak{B}\) is replaced by any Banach space \(\mathfrak{B}'\) of sequences \(\xi=\{\xi_j\}_1^\infty\) having the property: if \(\xi=\{\xi_j\}_1^\infty\in\mathfrak{B}'\), then the vectors \(\xi'=\{0,\xi_1,\ldots\}\) and \(\xi''=\{\xi_{j+1}\}_1^\infty\) also belong to \(\mathfrak{B}'\), and moreover \(|\xi'|\leq |\xi|\) and \(|\xi''|\leq |\xi|\).
§ 2.1. Let \(x\) be an arbitrary element of \(R\) and let \(\Lambda_j^+\) \((j=0,1,\ldots,\omega^+;\ \omega^+\leq \infty)\), \(\Lambda_j^-\) \((j=1,2,\ldots,\omega^-;\ \omega^-\leq \infty)\) be certain distinct connected components of the regularity domain of the element \(x\), with \(\lambda=\infty\in\Lambda_0^+\).
Introduce the following notation: \(\mathfrak{K}^+\) is the linear span of the system \(x^n\), \((x-\lambda_j^+e)^{-n}\) \((\lambda_j^+\in\Lambda_j^+;\ j=1,2,\ldots,\omega^+;\ n=0,1,\ldots)\); \(\mathfrak{K}^-\) is the linear span of the system \((x-\lambda_j^-e)^{-n}\) \((\lambda_j^-\in\Lambda_j^-;\ j=1,2,\ldots,\omega^-;\ n=0,1,\ldots)\); \(\mathfrak{K}\) is the linear span of the union of both systems; \(\mathfrak{R}, \mathfrak{R}^+, \mathfrak{R}^-\) are the closures (in the norm of the ring \(R\)) respectively of the rings \(\mathfrak{K}, \mathfrak{K}^+\), and \(\mathfrak{K}^-\); \(\Lambda^\pm=\bigcup_j \Lambda_j^\pm\) and \(S\) is the complement of the domain \(\Lambda^+\cup\Lambda^-\).
Obviously, \(\mathfrak{R}, \mathfrak{R}^{\pm}\) are normed subrings of the ring \(R\), and these subrings do not depend on the choice of the numbers \(\lambda_j^\pm\) in the corresponding components \(\Lambda_j^\pm\). The bicompacts of maximal ideals of the rings \(\mathfrak{R}, \mathfrak{R}^+, \mathfrak{R}^-\) are homeomorphic respectively to the sets \(S, S\cup\Lambda^-\), \(S\cup\Lambda^+\). As before, by \(z(\xi)\) we shall denote the function of the element \(z\) on the bicompact of maximal ideals of the ring.
If the set \(S\) coincides with the spectrum of the element \(x\) in \(R\), then the spectrum of every element \(z\in\mathfrak{R}\) is the same in the rings \(R\) and \(\mathfrak{R}\).
Denote by \(\varkappa_j^\pm(r)\) \((j=1,2,\ldots,\omega_j^\pm)\) the functional defined on all invertible elements \(r\in\mathfrak{K}\) \((r(\xi)\ne 0;\ \xi\in S)\) and equal to the difference between the number of poles and the number of zeros in \(\Lambda_j^\pm\) of the function \(r(\xi)\). The functionals \(\varkappa_j^\pm(r)\) extend by continuity to the set \(O\) of all invertible elements of \(\mathfrak{R}\) and have the following properties: 1) \(\varkappa_j^\pm(a)\) is an integer for every \(a\in O\); 2) \(\varkappa_j^\pm(a_1a_2)=\varkappa_j^\pm(a_1)+\varkappa_j^\pm(a_2)\) \((a_1,a_2\in O)\).
Theorem 4. The general form of every functional \(\varkappa(a)\), defined on \(O\) and possessing properties 1) and 2), is given by the formula
\[ \varkappa(a)=\sum_j c_j^+\varkappa_j^+(a)+\sum_j c_j^-\varkappa_j^-(a)\quad (a\in O), \tag{6} \]
where \(c_j^\pm\) are arbitrary integers.
The sums on the right-hand side of equality (6) are meaningful, since for each \(a\in O\) only a finite number of the functionals \(\varkappa_j^\pm\) may be nonzero.
Theorem 5. In order that every element \(a\in\mathfrak{R}\) satisfying the condition \(a(\xi)\ne 0\) \((\xi\in S)\) be representable in the form
\[ a=a_-(x-\lambda^-e)^\varkappa a_+ \qquad \left(\lambda^-\in\Lambda^-;\quad \varkappa=\varkappa(a)=\sum_j \varkappa_j^-(a)\right), \tag{7} \]
* An operator generated in one of the indicated spaces \(B\) by a matrix \(\|a_{j-k}\|\), \(j,k=1,2,\ldots\), satisfying the condition \(\sum |\alpha_j|<\infty\), can obviously be represented in the form (5).
where \(a_\pm\) are invertible elements, respectively, of the rings \(\mathcal R^\pm\); it is necessary and sufficient that every element \(z\in\mathcal R\) be representable in the form
\[ z=z_+ + z_- \qquad (z_+\in\mathcal R^+,\ z_-\in\mathcal R^-). \tag{8} \]
This theorem is a generalization of Theorem 2 and of some propositions from the theory of the Hilbert boundary-value problem \({}^{(3)}\). Its proof is based on a theorem of G. E. Shilov \(\left({}^{2},\S 13\right)\).
§ 2.2. Suppose that in the ring \(R\) there exists an idempotent element \(p\) \((p^2=p;\ p\ne e)\) with respect to which \(x\) has the following properties:
\[ pxp=xp,\qquad p(x-\lambda_j^+e)^{-1}p=(x-\lambda_j^+e)^{-1}p \quad (j=1,2,\ldots,\omega^+); \]
\[ qxq\ne xq,\qquad q(x-\lambda_j^-e)^{-1}q=(x-\lambda_j^-e)^{-1}q \quad (q=e-p;\ j=1,2,\ldots,\omega^-). \]
Under these assumptions the following holds.
Theorem 6. Let \(b,c\in\mathcal R\). If
\[ b(\zeta)\ne 0,\qquad c(\zeta)\ne 0 \qquad (\zeta\in S), \tag{9} \]
then for \(\varkappa(a)>0\) \(\left(a=bc^{-1},\ \varkappa(a)=\sum_j \varkappa_j^-(a)\right)\) the element \(bp+cq\) \((pb+qc)\) is strictly left-invertible in \(R\); for \(\varkappa(a)<0\) it is strictly right-invertible in \(R\); and for \(\varkappa(a)=0\) the element \(bp+cq\) \((pb+qc)\) has a two-sided inverse in \(R\). If condition (8) is satisfied in the ring \(\mathcal R\), then in all three cases
\[ (bp+cq)^{(-1)}=(a_+p+a_-^{-1}q)\bigl[(x-\lambda^-e)^{-\varkappa(a)}p+q\bigr]c^{-1}a_-^{-1} \qquad (\lambda^-\in\Lambda^-), \]
\[ (pb+qc)^{(-1)}=c^{-1}a_+^{-1}\bigl[p(x-\lambda^-e)^{-\varkappa(a)}+q\bigr](pa_-^{-1}+qa_+) \qquad (\lambda^-\in\Lambda^-), \]
where \(a_\pm\) are determined from equality (7), if one sets \(a=bc^{-1}\). If at no point \(\lambda\in S\) are the elements \((x-\lambda e)p+q\), \(\bigl[(x-\lambda^-e)^{-1}-(\lambda-\lambda^-)^{-1}e\bigr]q+p\) right-invertible in \(R\), and the elements \(q(x-\lambda e)+p\), \(p\bigl[(x-\lambda^-e)^{-1}-(\lambda-\lambda^-)^{-1}e\bigr]+q\) have no left inverses in \(R\), then (9) is a necessary condition for the element \(bp+cq\) \((pb+qc)\) to be invertible in \(R\) from at least one side.
For lack of space we omit here the statement of the theorem analogous to Theorem 3.
§ 2.3. As a consequence of the results of §§ 2.1 and 2.2 one can obtain a generalization of the theorems for paired systems with Toeplitz matrices of type (1) and their transposes \(\left({}^{4},\S 7\right)\). In addition, all the fundamental assertions on singular integral equations with discontinuous coefficients on multiply connected closed contours in \(L_p\) \((1<p<\infty)\) \({}^{(5-8)}\) follow from these results.
The results presented in this communication are generalized to the case of matrices composed of elements of the classes considered.
The author expresses his gratitude to M. G. Krein and G. E. Shilov for valuable discussion of the results of this communication.
Institute of Physics and Mathematics
Academy of Sciences of the USSR
Received
9 III 1962
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