MATHEMATICS

G. R. BELITSKII

ON AUTOMORPHISMS OF THE ORDER STRUCTURE ON THE SET OF MATRIX NORMS

(Presented by Academician S. N. Bernstein on 21 V 1965)

Let \(\mathfrak M_p\) denote the ring of real square matrices of order \(p\), and let \(\mathfrak N_p\) denote the ordered (partially) set of norms\(^*\) in \(\mathfrak M_p\). The structure \(\mathfrak N_p\) has already been considered in the works \((^{1,2})\). Here we shall undertake its further study.

We shall call an automorphism of the structure \(\mathfrak N_p\) a mapping of the set \(\mathfrak N_p\) onto itself that preserves the order relation. An example of an automorphism is the mapping \(\varphi_u\), which assigns to each norm \(n(A)\) the norm \(\varphi_u n(A)=n(UAU^{-1})\), and also the mapping \(\varphi_u^*\): \(\varphi_u^* n(A)=n(UA'U^{-1})\), where \(U\) is any nonsingular matrix and the prime denotes transposition.

Theorem. The structure \(\mathfrak N_p\) has no automorphisms other than \(\varphi_u\) and \(\varphi_u^*\).

Thus we have a complete description of the automorphism group of the structure \(\mathfrak N_p\).

Before outlining the scheme of the proof, let us introduce one auxiliary notion, which will play a very important role in what follows.

Let \(\pi \subseteq \mathfrak M_p\) be any subring. Denote by \(\mathfrak N_p(\pi)\) the set of all functionals on \(\pi\), each of which is the lower bound of some chain of norms on \(\pi\). From the results of \((^2)\) it follows that \(\mathfrak N_p(\mathfrak M_p)=\mathfrak N_p\).

Definition. Let \(\pi \subseteq \mathfrak M_p\) be a subring and \(\nu \in \mathfrak N_p(\pi)\). The generalized norm \(n(A;\pi,\nu)\) \((A\in\mathfrak M_p)\) is the functional of \(A\) equal to \(\nu\) on \(\pi\) and to infinity outside \(\pi\). We shall denote the totality of all generalized norms by \(\overline{\mathfrak N}_p\). The set \(\overline{\mathfrak N}_p\) is considered ordered by the same principle as \(\mathfrak N_p\).

Thus, generalized norms formally have all the properties of matrix norms, except, perhaps, positivity and finiteness on each matrix.

Lemma 1. Let \(\varphi\) be any automorphism of the structure \(\mathfrak N_p\). There exists, and moreover uniquely, an automorphism \(\overline{\varphi}\) of the structure \(\overline{\mathfrak N}_p\) that coincides with \(\varphi\) on \(\mathfrak N_p\).

In what follows we shall identify the automorphism \(\overline{\varphi}\) with the automorphism \(\varphi\). By virtue of Lemma 1, our theorem is equivalent to the analogous assertion for the structure \(\overline{\mathfrak N}_p\). We shall prove precisely this assertion.

Consider the structure, ordered by inclusion, of the set of subrings of the ring \(\mathfrak M_p\). The automorphism \(\varphi\) (henceforth fixed) induces a certain mapping \(\Phi\) of this structure onto itself. Namely, if
\[ \varphi n(A;\pi,\nu) \equiv n(A;\pi',\nu'), \]
then put \(\Phi(\pi,\nu)=\pi'\). The images of one and the same subring \(\pi\) may be different subrings \(\pi'\), depending on the “parameter” \(\nu\). The mapping \(\Phi\), in a certain sense, preserves the relation—

\(^*\) Recall that a norm in a ring is, by definition, “submultiplicative”:
\[ \|AB\|\leq \|A\|\,\|B\|. \]
The order relation is defined in the natural way: if \(\nu_1,\nu_2\in\mathfrak N_p\), then
\[ \nu_1<\nu_2 \Longleftrightarrow \nu_1(A)\leq \nu_2(A),\quad (A\in\mathfrak M_p)\ \&\ \nu_1(A)\ne\nu_2(A). \]

the order of the subring structure. Namely, if \(\pi_1 \subseteq \pi_2\) and \(\nu_1(A) \geq \nu_2(A)\), \(A \in \pi_1\), then \(\Phi(\pi_1,\nu_1) \subseteq \Phi(\pi_2,\nu_2)\). The minimal elements of the subring structure will be the one-dimensional subrings \(G_X\) with one generator \(X\) such that \(X^2=\mu X\), and only these. Therefore it is clear that the set \(G=\bigcup_X G_X\) is invariant with respect to \(\Phi\). Consequently, the set \(\overline{\mathfrak N}_p^{(0)}\) of all norms of the form \(n(A;G_X,\nu)\) is invariant with respect to the automorphism \(\varphi\). In fact, one can prove somewhat more.

Lemma 2. The set \(*\,\mathfrak N_X\) of all generalized norms of the form \(n(A;G_X,\nu)\) for fixed \(X\) is carried by the automorphism \(\varphi\) into the set \(\mathfrak N_Y\) of all norms of the form \(n(A;G_Y,\nu)\) with some \(Y=Y(\varphi,X)\).

In other words, \(\Phi(G_X,\nu)=G_Y\) does not depend on \(\nu\). Thus, to each matrix \(X\in G\), up to a factor, there corresponds a matrix \(Y=\Phi_0(X)\in G\). Imposing on this correspondence the requirement of homogeneity: \(\Phi_0(\lambda X)=\lambda\Phi_0(X)\), we obtain a single-valued operator \(\Phi_0\) on the set \(G\). It is easy to see that if a subring \(\pi\subset G\), then

\[ \Phi(\pi,\nu)=\bigcup \Phi_0(X). \]

Lemma 3. If two norms \(n_1,n_2\in\overline{\mathfrak N}_p\) coincide on the matrix \(X\in G\), then their images coincide on the matrix \(\Phi_0(X)\).

Proof. It is clearly sufficient to prove the lemma for the case when \(n_2=n(A;G_X,\nu)\), \(\nu=n_1(X)\). In this case \(n_2>n_1\), and therefore \(\varphi n_2>\varphi n_1\equiv n_1'\). Suppose that \(\varphi n_2(\Phi_0(X))>n_1'(\Phi_0(X))\equiv\nu'\). Consider the norm \(n_2'=n(A;G_{\Phi_0(X)},\nu')\). Then \(\varphi n_2>n_2'>n_1'\). Applying the inverse automorphism, we obtain: \(n_2>\varphi^{-1}n_2'>n_1\). At the same time certainly \(n_2(X)>\varphi^{-1}n_2'(X)\geq n_1(X)\), contrary to the condition.

From Lemma 2 it follows that \(\varphi n(A;G_X,\nu)=n(A;G_{\Phi_0(X)},\nu')\), where \(\nu'=f(X,\nu)\geq 0\) is some monotone function in \(\nu\), finite everywhere. Moreover, the mapping \(\Phi\) generates a certain automorphism of the structure of subrings lying in \(G\). Investigation of this automorphism shows that it leaves invariant the set \(\mathfrak M_p^{(1)}\subset G\) of matrices of rank one. Using Lemma 3, we arrive at the following result.

Lemma 4. There exists such a nonsingular matrix \(U\) that either

\[ \Phi_0(T)=\varepsilon(T)U^{-1}TU \qquad (T\in\mathfrak M_p^{(1)}),\ |\varepsilon(T)|=1, \]

or

\[ \Phi_0(T)=\varepsilon(T)U^{-1}T' U \qquad (T\in\mathfrak M_p^{(1)}), \]

where \(\varepsilon(T)\) is a scalar, \(|\varepsilon(T)|=1\). Moreover, \(f(\nu,T)=\nu\) for \(T\in\mathfrak M_p^{(1)}\).

From Lemma 4 there immediately follows the validity of our theorem for the subset of norms from \(\mathfrak N_p^{(0)}\) that are finite on matrices of rank one. This, in turn, entails its validity for a fairly broad class of norms, first of all for those \(n\in\overline{\mathfrak N}_p\) which are the structural lower bound of norms from the indicated subset. Further, we shall call a norm \(n\in\overline{\mathfrak N}_p\) minimal with respect to a set \(M\subset\mathfrak N_p\) if from the conditions: a) \(n_1\in\mathfrak N_p\), b) \(n_1(A)=n(A)\) \((A\in M)\), it follows that c) \(n_1>n\). It is not difficult now to understand that our theorem is valid for norms minimal with respect to the set \(\mathfrak M_p^{(1)}\) and, in particular (see (1)), for operator norms.

To complete the proof of the theorem, consider the subring \(D_V\) of all matrices of the form \(Y=VXV^{-1}\), where \(X\) is a diagonal matrix and \(V\) is a fixed nonsingular matrix.

Lemma 5. The domain of finiteness of the norm \(\varphi n(A;D_V,\nu)\) is the subring \(D_{U^{-1}V}\) (or \(D_{U^{-1}V'}\)).

This assertion follows in an obvious way from the fact that any norm \(n(A;D_V,\nu)\) is majorized and minorized by norms with the same domain of finiteness \(D_V\), for which the validity of our theorem has already been established—

* It is a chain.

the theorem. Namely,

\[ n_1 \equiv n(A; D_\nu, \nu_1) = \inf_{T \in D_\nu \cap \mathfrak{M}_p^{(1)}} n(A; G_T, \nu) \ge \]

\[ \ge n(A; D_\nu, \nu) \ge n_2(A) \equiv n(A; D_\nu, \nu_2), \]

where \(\nu_2(A) = r(A)\) is the spectral radius of the matrix \(A \in D_\nu\). It is not difficult to verify that \(n_2\) is minimal with respect to the set \(\mathfrak{M}_p^{(1)}\), so that the theorem is valid for this norm.

Corollary. If two norms \(n_1, n_2 \in \overline{\mathfrak{M}}_p\) coincide on the subring \(D_\nu\), then their images coincide on the subring \(D_{U^{-1}\nu}\) (or on \(D_{U^{-1}\nu}\)).

The proof of this assertion does not differ in any way from the proof of Lemma 3.

From Lemma 5 and its corollary it follows that our theorem is valid for norms of the form \(n(A; D_\nu, \nu)\).

Finally, let \(n \in \mathfrak{M}_p\). From the equality \(n(A) = \inf_\nu n(A; D_\nu, \nu_\nu)\), where \(\nu_\nu(A)=n(A)\) \((A \in D_\nu)\), the validity of the theorem for the norm \(n \in \mathfrak{M}_p\) follows, and consequently also for all norms in \(\mathfrak{M}_p\).

Received
20 V 1965

CITED LITERATURE

1. Yu. I. Lyubich, UMN, 18, No. 4, 161 (1963).
2. G. R. Belitskii, DAN, 151, No. 1, 9 (1963).