A. S. MARKUS

ON THE EIGENVALUES AND SINGULAR VALUES OF THE SUM AND PRODUCT OF LINEAR OPERATORS

(Presented by Academician P. S. Aleksandrov on 5 IV 1962)

V. B. Lidskii (¹) proved a theorem establishing, in a visual geometric form, a connection between the eigenvalues of the sum of Hermitian matrices and the eigenvalues of the summands, and also solved the analogous problem for the product of positive definite matrices*.

In the present note we give several propositions on the singular values of matrices (Sec. 2), analogous to the theorems of V. B. Lidskii, and also extensions of V. B. Lidskii’s theorems and of the indicated propositions to the infinite-dimensional case (Secs. 3—5). The proofs of the theorems given below are based on certain assertions of a geometric character (Sec. 1), established with the aid of the theorem of M. G. Krein and D. P. Milman (³) on the extreme points of a convex bicompact set, and also on the ideas of Weyland’s work (⁴), which indicated a new proof of V. B. Lidskii’s theorem on the eigenvalues of the sum of Hermitian matrices.

1. Let \(R^n\) be an \(n\)-dimensional Euclidean space, \(c_0\) (respectively \(l_1\)) the space of real sequences tending to zero (respectively absolutely summable), \(\alpha=\{\alpha_j\}_1^\infty\), \(\hat c_0\) and \(\hat l_1\) the same spaces, but consisting of sequences \(\alpha=\{\alpha_j\}_{-\infty}^{\infty'}\), where the prime denotes omission of the index \(j=0\).

If \(\alpha=\{\alpha_j\}\) and \(\beta=\{\beta_j\}\) are vectors of the space \(R^n\) (or \(c_0\), or \(\hat c_0\)) and

\[ \sup_{j_1<\cdots<j_k}\sum_{m=1}^{k}\alpha_{j_m} \le \sup_{j_1<\cdots<j_k}\sum_{m=1}^{k}\beta_{j_m} \quad (k=1,2,\ldots), \]

then we shall write \(\alpha\ll\beta\). If \(\alpha,\beta\in\hat l_1\), \(\alpha\ll\beta\), \(-\alpha\ll-\beta\), and

\[ \sum_{j=-\infty}^{\infty'} \alpha_j = \sum_{j=-\infty}^{\infty'} \beta_j, \]

then we shall write \(\alpha<\beta\).

Lemma 1. Let \(\beta\in R^n\) and \(\beta_j\ge 0\) \((j=1,2,\ldots,n)\). For a vector \(\alpha\in R^n\) the condition
\[ (|\alpha_1|,|\alpha_2|,\ldots,|\alpha_n|)\ll(\beta_1,\beta_2,\ldots,\beta_n) \]
is fulfilled if and only if \(\alpha\) is contained in the convex hull of all possible vectors of the form
\[ (\varepsilon_1\beta_{j_1},\varepsilon_2\beta_{j_2},\ldots,\varepsilon_n\beta_{j_n}), \]
where \(\varepsilon_k=\pm 1\) and \(j_1,j_2,\ldots,j_n\) is an arbitrary permutation of the numbers \(1,2,\ldots,n\).

Lemma 2. Let \(\beta\in c_0\) and \(\beta_j\ge 0\) \((j=1,2,\ldots)\). For a real vector \(\alpha=\{\alpha_j\}_1^\infty\) the condition
\[ (|\alpha_1|,|\alpha_2|,\ldots)\ll(\beta_1,\beta_2,\ldots) \]
is fulfilled if and only if \(\alpha\) is contained in the convex hull, closed in the norm of \(c_0\), of all possible vectors of the form

\[ (\varepsilon_1\beta_{j_1},\varepsilon_2\beta_{j_2},\ldots), \tag{1} \]

where \(\varepsilon_k=\pm 1\) and \(j_1,j_2,\ldots\) is an arbitrary permutation of the natural sequence.

Remark 1. If \(\beta\in l_p\) \((p\ge 1)\), then the convex hulls of all possible vectors of the form (1), closed in the norm of \(c_0\) and in the norm of \(l_p\), coincide.

Lemma 3. If \(\beta\in\hat l_1\) (respectively \(\hat c_0\)), then the convex hull, closed in the norm of \(\hat l_1\) (respectively \(\hat c_0\)), of all possible vectors obtained

* These problems first arose in the investigations of I. M. Gelfand and M. A. Naimark on the theory of representations of groups; their solution, based on group methods, was indicated by F. A. Berezin and I. M. Gelfand (²).

by permutations of the coordinates of the vector \(\beta\), coincides with the set of all vectors \(\alpha\) such that \(\alpha \prec \beta\) (respectively, \(\alpha \ll \beta\) and \(-\alpha \ll -\beta\)).

Remark 2. If \(\beta \in \hat l_p\) \((p>1)\), then the convex closures, closed in the norm \(\hat c_0\) and in the norm \(\hat l_p\), of all possible vectors obtained by permutations of the coordinates of the vector \(\beta\), coincide.

This lemma (in the part concerning \(\hat l_1\)) is an infinite-dimensional analogue of a proposition of Rado \({}^{(5)}\) (which can also be obtained by combining earlier results of Hardy, Littlewood, and Pólya \({}^{(6)}\), p. 66, and Birkhoff \({}^{(7)}\)).

1. Let \(A\) be a complex matrix of order \(n\). By \(s_j(A)\) \((j=1,2,\ldots,n)\) we denote the singular numbers of the matrix \(A\), i.e. the square roots, arranged in decreasing order, of the eigenvalues of the matrix \(A^*A\).

The following proposition is established with the aid of certain results of Wielandt \({}^{(4)}\) and Mirsky \({}^{(8)}\) and Lemma 1.

Theorem 1. If \(A\) and \(B\) are matrices of order \(n\) and \(C=A+B\), then the vector
\[ (s_1(C),\,s_2(C),\,\ldots,\,s_n(C)) \]
is contained in the convex hull of all possible vectors of the form
\[ (s_1(B)+\varepsilon_1 s_{j_1}(A),\,s_2(B)+\varepsilon_2 s_{j_2}(A),\,\ldots,\,s_n(B)+\varepsilon_n s_{j_n}(A)), \]
where \(\varepsilon_k=\pm1\), and \(j_1,j_2,\ldots,j_n\) is an arbitrary permutation of the numbers \(1,2,\ldots,n\).

Corollary 1. If, under the hypotheses of Theorem 1,
\[ s_1(A)+s_2(A)<s_k(B)-s_{k+1}(B)\qquad (k=1,2,\ldots,n-1), \]
then all the numbers \(s_j(C)\) \((j=1,2,\ldots,n)\) are distinct.

Theorem 2. If \(A\) and \(B\) are nonsingular matrices of order \(n\) and \(C=AB\), then the vector
\[ (\ln s_1(C),\,\ln s_2(C),\,\ldots,\,\ln s_n(C)) \]
is contained in the convex hull of all possible vectors of the form
\[ (\ln s_1(B)+\ln s_{j_1}(A),\,\ln s_2(B)+\ln s_{j_2}(A),\,\ldots,\,\ln s_n(B)+\ln s_{j_n}(A)), \]
where \(j_1,j_2,\ldots,j_n\) is an arbitrary permutation of the numbers \(1,2,\ldots,n\).

This theorem follows from results of Amir-Moéz \({}^{(9)}\), Theorem 3.10, and Rado \({}^{(5)}\).

1. Following \({}^{(10)}\), by \(\mathfrak S_\infty\) we denote the set of all linear completely continuous operators acting in a separable Hilbert space \(\mathfrak H\), and by \(\mathfrak S_1\) the subset of \(\mathfrak S_\infty\) consisting of operators with finite trace.

If \(A\) is a self-adjoint completely continuous operator, then by
\[ \lambda(A)=\{\lambda_j(A)\}_{-\infty}^{\infty} \]
we shall denote the complete system of eigenvalues of the operator \(A\), in which they are numbered (with multiplicities taken into account) by indices from \(-\infty\) to \(\infty\), with the omission of the index \(j=0\), and moreover
\[ \lambda_j(A)\ge 0,\quad \lambda_{-j}(A)\le 0,\quad \lambda_j(A)\ge \lambda_{j+1}(A),\quad \lambda_{-j}(A)\le \lambda_{-j-1}(A)\quad (j=1,2,\ldots)^{*}. \]

Lemma 4. If \(H\in\mathfrak S_\infty\) and \(T\in\mathfrak S_1\) are self-adjoint operators, then
\[ \sum_{j=-\infty}^{\infty}\bigl(\lambda_j(H+T)-\lambda_j(H)\bigr)=\operatorname{Sp} T, \]
and the series converges absolutely.

With the aid of Lemmas 3 and 4 one proves the following proposition, which is a generalization of a theorem of V. B. Lidskii \({}^{(1)}\).

Theorem 3. Let \(A\) be a self-adjoint operator in \(\mathfrak S_\infty\) (respectively \(\mathfrak S_1\)), \(B\) a self-adjoint operator in \(\mathfrak S_\infty\), and \(C=A+B\). Then,

* The term “complete” has here not quite its usual meaning, since if the operator \(A\) has an infinite number of both positive and negative eigenvalues, then the sequence \(\{\lambda_j(A)\}_{-\infty}^{\infty}\) will consist only of them, and, consequently, the eigenvalues equal to zero will not enter it, even if such exist; if the operator \(A\) has a finite number \(n\) \((\ge 0)\) of positive (negative) eigenvalues, then we put \(\lambda_{n+j}(A)=0,\ j=1,2,\ldots\) \((\lambda_{-n-j}(A)=0,\ j=1,2,\ldots)\), irrespective of whether the number 0 is an eigenvalue of the operator \(A\) or not.

the vector \(\lambda(C)-\lambda(B)\) is contained in the convex closed (in the norm \(\hat c_0\) (respectively \(\hat l_1\)) hull of all possible vectors obtained by permuting the coordinates of the vector \(\lambda(A)\).

1. If \(A\in \mathfrak S_\infty\), then by \(s(A)=\{s_i(A)\}_1^\infty\) is denoted the sequence of singular numbers of the operator \(A\), i.e., the sequence of eigenvalues of the operator \((A^*A)^{1/2}\), numbered in decreasing order with their multiplicities taken into account.

The following theorem, proved with the aid of Lemma 2, is a generalization of Theorem 1, and also of one result of Fan Ky ((\(^{11}\)), Theorem 5).

Theorem 4. If \(A\) and \(B\) are completely continuous operators and \(C=A+B\), then the vector \(s(C)-s(B)\) is contained in the convex closed (in the norm \(c_0\)) hull of all possible vectors of the form \((\varepsilon_1 s_{j_1}(A), \varepsilon_2 s_{j_2}(A),\ldots)\), where \(\varepsilon_k=\pm 1\), and \(j_1,j_2,\ldots\) is any permutation of the natural sequence.

With the aid of Fan Ky’s theorem ((\(^{11}\)), Theorem 4) one obtains

Corollary 2. Let \(\|A\|\) be some unitarily invariant norm and \(\Phi(s_1,s_2,\ldots)\) the symmetric norming function generating it (in the sense of Neumann and Schatten (\(^{12}\))). Then, for any \(A,B\in\mathfrak S_\infty\),
\[ \|A-B\|\geq \Phi(s_1(A)-s_1(B),\,s_2(A)-s_2(B),\ldots). \]

For the finite-dimensional case this assertion was obtained by Mirsky (\(^{8}\)).

1. Theorem 5. If \(A\) and \(B\) are completely continuous operators and \(C=AB\), then for every natural \(k\) and any distinct natural numbers \(j_1,j_2,\ldots,j_k\)
   \[ \prod_{n=1}^{k}s_{j_n}(C)\leq \prod_{n=1}^{k}s_n(A)\prod_{n=1}^{k}s_{j_n}(B). \]

This theorem is a generalization of Horn’s theorem (\(^{13}\)).

Theorem 2 admits a generalization, for the formulation of which we shall need the following notation. If \(T\in \mathfrak S_\infty\) and \(A=I+T\), put
\[ s_j(A)=\bigl(1+\lambda_j(T+T^*+T^*T)\bigr)^{1/2}\quad (j=\pm1,\pm2,\ldots). \]

Theorem 6. Let \(T_1\in\mathfrak S_\infty\) (respectively \(T_1\in\mathfrak S_1\)) and \(T_2\in\mathfrak S_\infty\), the operators \(A=I+T_1\) and \(B=I+T_2\) vanish only at zero, and let \(C=AB\). Then the vector \(\{\ln s_j(C)-\ln s_j(B)\}_{-\infty}^{\infty}\) is contained in the convex closed (in the norm \(\hat c_0\) (respectively \(\hat l_1\)) hull of all possible vectors obtained by permuting the coordinates of the vector \(\{\ln s_j(A)\}_{-\infty}^{\infty}\).

There also hold assertions, analogous to Theorems 5 and 6, about the eigenvalues of the product of positive operators (the second of them is an infinite-dimensional analogue of Theorem 4 of V. B. Lidskii (\(^{1}\))).

Let us note in conclusion that, in the assertions of all the theorems presented, one may interchange the operators \(A\) and \(B\) (for Theorems 1, 3, and 4 this is obvious, while for Theorems 2, 5, and 6 it follows from the coincidence of the singular numbers of the operators \(A\) and \(A^*\)).

The author expresses his gratitude to I. Ts. Gokhberg for valuable discussions of the questions considered here.

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

Received
4 IV 1962

CITED LITERATURE

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4. H. Wielandt, Proc. Am. Math. Soc., 6, No. 1, 106 (1955).
5. R. Rado, J. London Math. Soc., 27, 1 (1952).
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7. G. Birkhoff, Revista Universidad Nacional Tucumán, ser. A, 5, 147 (1946).
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9. A. Amir-Moéz, Duke Math. J., 23, 463 (1956).
10. I. Ts. Gokhberg, M. G. Krein, DAN, 137, No. 5, 1034 (1961).
11. Ky Fan, Proc. Nat. Acad. Sci. USA, 37, 760 (1951).
12. R. Schatten, Norm Ideals of Completely Continuous Operators, Berlin—Göttingen—Heidelberg, 1960.
13. A. Horn, Proc. Nat. Acad. Sci. USA, 36, 374 (1950).