MATHEMATICS

Yu. P. GINZBURG

ON PROJECTION IN A HILBERT SPACE WITH A BILINEAR METRIC

(Presented by Academician L. S. Pontryagin on 21 III 1961)

Let, in a Hilbert space \(\mathfrak H\) with scalar product \((f,g)\), a continuous bilinear (generally speaking, not Hermitian-symmetric) functional be given,

\[ [f,g]=(Gf,g). \]

We shall call the bounded linear operator \(G\) the Gram operator of the functional \([f,g]\) in the space \(\mathfrak H\).

Let \(\mathfrak M\) be a linear manifold in \(\mathfrak H\). A vector \(g\in\mathfrak M\) will be called a (left) \(G\)-projection of the vector \(f\) onto \(\mathfrak M\) if \([f-g,h]=0\) for all \(h\in\mathfrak M\). An operator \(Q_{\mathfrak M}\) (generally speaking, multivalued) will be called a \(G\)-projector onto \(\mathfrak M\) if to each \(f\in\mathfrak H\) it assigns the set (possibly empty) of \(G\)-projections of \(f\) onto \(\mathfrak M\). Some sufficient conditions for the existence of \(G\)-projections in the case of Hermitian-symmetric \([f,g]\) were obtained by R. Nevanlinna \((^1)\) and I. S. Louhivaara \((^2)\). In the case where the form \([f,f]\) has a finite number of negative squares, this question was considered earlier by L. S. Pontryagin \((^3)\), and then by I. S. Iokhvidov and M. G. Krein \((^{4,5})\).

In the present note a formula is given which relates the \(G\)-projector onto a linear manifold \(\mathfrak M\) to an operator \(T\) whose range coincides with \(\mathfrak M\), and a number of criteria for the existence and uniqueness of \(G\)-projections are formulated, from which, in particular, the aforementioned results of Nevanlinna and Louhivaara follow. In addition, for the special case \(G=J\) \((J^2=I,\ J^*=J)\) we give a complete description of those definite (positive and negative \((^4)\)) subspaces \(\mathfrak M\) for which the operator \(Q_{\mathfrak M}\) is defined on all of \(\mathfrak H\).

1. Let \(A\) be an arbitrary linear operator in \(\mathfrak H\). By \(A^{-1}\) we shall henceforth mean the operator (generally speaking, multivalued) which assigns to each \(f\in\mathfrak H\) the set (possibly empty) of all its preimages \(h\) \((Ah=f)\).

Theorem 1. If \(\mathfrak M\) is the range of a bounded linear operator \(T\), then

\[ Q_{\mathfrak M}=T\,(T^*GT)^{-1}T^*G. \]

If \(\mathfrak M\) is a subspace, then, putting \(T=P_{\mathfrak M}\), where \(P_{\mathfrak M}\) is the usual orthogonal projector onto \(\mathfrak M\), we obtain

\[ Q_{\mathfrak M}=G_{\mathfrak M}^{-1}P_{\mathfrak M}G \]

(here \(G_{\mathfrak M}=P_{\mathfrak M}GP_{\mathfrak M}\) is the Gram operator of the functional \([f,g]\) in the subspace \(\mathfrak M\)).

Hence the following assertions follow directly:

\(1^\circ\). In order that each vector of \(\mathfrak H\) have no more than one \(G\)-projection onto the subspace \(\mathfrak M\), it is necessary and sufficient that on \(\mathfrak M\)

the operator \(G_{\mathfrak M}\) not vanish (such a subspace \(\mathfrak M\) will be called nondegenerate).

\(2^\circ\). In order that the vector \(f_0\) have a \(G\)-projection onto the subspace \(\mathfrak M\), it is necessary and sufficient that

\[ P_{\mathfrak M}Gf_0\in \mathfrak R(G_{\mathfrak M}) \]

(by the symbol \(\mathfrak R(A)\) we denote the range of the operator \(A\)).

\(3^\circ\). Every vector in \(\mathfrak H\) has at least one \(G\)-projection onto the subspace \(\mathfrak M\) if and only if

\[ \mathfrak R(P_{\mathfrak M}G)=\mathfrak R(G_{\mathfrak M}). \]

Below we formulate assertions \(1^\circ\) and \(2^\circ\) in another form. To this end, in the subspace \(\mathfrak M\subseteq \mathfrak H\) consider the seminorm \((^6)\)
\[ |f|_{\mathfrak M}=\|G_{\mathfrak M}f\|=(G_{\mathfrak M}G_{\mathfrak M}^{*}f,f)^{1/2} \]
(in particular, \(|f|_{\mathfrak H}=\|G^{*}f\|\)). We shall call the subspace \(\mathfrak M\) relatively regular if the seminorms \(|f|_{\mathfrak M}\) and \(|f|_{\mathfrak H}\) are equivalent on \(\mathfrak M\), which in the case under consideration reduces to the following: if \(|f_n|_{\mathfrak M}\to 0\) \((f_n\in\mathfrak M)\), then \(|f_n|_{\mathfrak H}\to 0\). We shall call the subspace \(\mathfrak M\) regular if on \(\mathfrak M\) the quantities \(|f_n|_{\mathfrak M}\) and \(\|f\|\) are equivalent. Obviously, every regular subspace is relatively regular. The converse is true only when \(\mathfrak H\) itself is regular. On the basis of Banach’s theorem on the inverse operator, the subspace \(\mathfrak M\) is regular if and only if \(|f|_{\mathfrak M}\) is a norm and in this norm \(\mathfrak M\) is complete, which for a definite \(\mathfrak M\) is equivalent to completeness in the norm \(|[f,f]|^{1/2}\).

Theorem 2. 1) In order that there exist a \(G\)-projection of the vector \(g_0\) onto the subspace \(\mathfrak M\), it is necessary and sufficient that the functional
\[ \varphi_{g_0}(f)=[g_0,f] \]
be continuous on \(\mathfrak M\) in the topology determined by the seminorm \(|f|_{\mathfrak M}\) (i.e., from \(\lim |f_n|_{\mathfrak M}=0\) \((f_n\in\mathfrak M)\) there follows the equality \(\lim \varphi_{g_0}(f_n)=0\)).

2) In order that every vector in \(\mathfrak H\) have at least one \(G\)-projection onto \(\mathfrak M\), it is necessary and sufficient that the subspace \(\mathfrak M\) be relatively regular.

In the proof of this theorem the following auxiliary proposition is used:

Let \(A\) and \(B\) be bounded linear operators in \(\mathfrak H\). Then, in order that

\[ \mathfrak R(A)\subseteq \mathfrak R(A), \]

it is necessary and sufficient that

\[ \|A^{*}h\|\leq \alpha\|\,\cdot\,\| \]

for some \(\alpha>0\) for all \(h\in\mathfrak H\).

1. Suppose now that the functional \([f,g]\) is Hermitian-symmetric, i.e., that \(G\) is a self-adjoint operator. In this case the condition for the existence of a \(G\)-projection of a vector onto a subspace can be formulated in another form. Let \(E_t^{(\mathfrak M)}\) \((a\leq t\leq b)\) be the resolution of the identity of the operator \(G_{\mathfrak M}\) \((E_a^{(\mathfrak M)}=0,\ E_b^{(\mathfrak M)}=P_{\mathfrak M})\). Then, on the basis of \(2^\circ\), for the existence of \(Q_{\mathfrak M}f_0\) it is necessary and sufficient that

\[ \int_a^b \frac{1}{t^2}\,d\bigl\|E_t^{(\mathfrak M)}Gf_0\bigr\|^2<\infty . \]

We have obtained a criterion for the existence of a \(G\)-projection, found by I. S. Louhivaara \((^2)\) and generalizing an earlier result of R. Nevanlinna \((^1)\).

Let us call the set \(\mathfrak N\) of vectors of \(\mathfrak H\) that are \(G\)-orthogonal to \(\mathfrak M \subset \mathfrak H\) the \(G\)-orthogonal complement of the subspace \(\mathfrak M\).

The following proposition follows easily from Theorem 2:

Theorem 3. In order that the equality

\[ \mathfrak M \dotplus \mathfrak N=\mathfrak H \]

hold, it is necessary and sufficient that \(\mathfrak M\) be nondegenerate and relatively regular. In this case \(\mathfrak N\) is also relatively regular, and if \(\mathfrak H\) is nondegenerate, then \(\mathfrak N^*\) is nondegenerate as well.

1. Let us consider the special case in which the Gram operator \(G\) of the functional \([f,g]\) has the form \(G=J=P_+-P_-\) (\(P_+,P_-\) are orthogonal projectors, \(P_+P_-=0\), \(P_+ + P_- = I\)). Put \(\mathfrak H_+=P_+\mathfrak H\), \(\mathfrak H_- = P_-\mathfrak H\).

Let \(\mathfrak M\) be a certain lineal in \(\mathfrak H\). If there exists a linear operator \(K\) mapping \(\mathfrak M^{(+)}=P_+\mathfrak M\) onto \(\mathfrak M^{(-)}=P_-\mathfrak M\) (or \(\mathfrak M^{(-)}\) onto \(\mathfrak M^{(+)}\)) and having the lineal \(\mathfrak M\) as its graph, then we shall call \(K\) the angular operator of the lineal \(\mathfrak M\) with respect to \(\mathfrak H_+\) (with respect to \(\mathfrak H_-\)) (cf. (7)).

Theorem 4. In order that a lineal \(\mathfrak M \subset \mathfrak H\) be a nonnegative subspace (4), it is necessary and sufficient that the angular operator \(K\) of the lineal \(\mathfrak M\) with respect to \(\mathfrak H_+\) exist and be nonexpanding on \(\mathfrak M^{(+)}\) \((\|K\| \leqslant 1)\). Moreover, \(\mathfrak M\) is positive if and only if \(K\) is a contraction on \(\mathfrak M^{(+)}\) \((\|Kf\|<\|f\|\) for \(0\ne f\in \mathfrak M^{(+)})\), null when \(K\) is isometric on \(\mathfrak M^{(+)}\) \((\|Kf\|=\|f\|,\ f\in \mathfrak M^{(+)})\), and regular when \(\|K\|<1\).

A theorem for nonpositive subspaces is formulated in a completely analogous way.

Since every subspace \(\mathfrak M \subset \mathfrak H\) admits a unique decomposition

\[ \mathfrak M=\mathfrak M_+ \oplus \mathfrak M_- \oplus \mathfrak M_0, \]

where \(\mathfrak M_+\), \(\mathfrak M_-\), \(\mathfrak M_0\) are, respectively, positive, negative, and null subspaces that are mutually orthogonal and \(J\)-orthogonal (see, for example, (*)), Theorem 4 makes it possible to give a description of all regular and irregular subspaces of the space \(\mathfrak H\) with indefinite \(J\)-metric.

From what has been said above it follows that in order that every nondegenerate subspace in \(\mathfrak H\) be regular, it is necessary and sufficient that the dimension of at least one of the subspaces \(\mathfrak H_+\) and \(\mathfrak H_-\) be finite (the sufficiency of this condition was proved in (*)).

Hence, and from Theorem 2, there follow results due to L. S. Pontryagin (³) concerning the existence of \(J\)-projections onto a nondegenerate subspace in the case \(\dim \mathfrak H_-<\infty\), and also the fact that, when \(\dim \mathfrak H_+=\dim \mathfrak H_-=\infty\), there exist in \(\mathfrak H\) nondegenerate subspaces \(\mathfrak M\) such that \(J\)-projections onto \(\mathfrak M\) do not exist for all vectors of \(\mathfrak H\). However, as is not difficult to see, the set of vectors that have a \(J\)-projection onto \(\mathfrak M\) is dense in \(\mathfrak H\). As examples show, for \(G\ne J\) this, generally speaking, is not so.

Let us note that Theorem 1 makes it possible to express, for example, the \(J\)-projector onto a maximal positive subspace \(\mathfrak M\) (⁵) in terms of the angular operator \(K\) of this subspace, namely,

\[ Q_{\mathfrak M}=(P_+ + K)(P_+ - K^*K)^{-1}(P_+ - K^*). \]

This theorem may also find application in the calculation of the coefficients of elementary divisors in the finite-dimensional case first obtained in

\[ \text{* After the present article had been submitted for publication, we became acquainted with the work }(^{10}), \]
from the results of which, in particular, a proposition close to Theorem 3 follows.

in the case of the multiplicative decomposition, due to V. P. Potapov,^8 of the analytic operator-function \(Y(\xi)\), which is two-sidedly \(J\)-nonexpanding^9 for \(|\xi|<1\). This application is based on the fact that if \(A\) is the leading coefficient of the Laurent expansion of the function \(Y(\xi)\) in a neighborhood of the pole \(\xi_0\) \((|\xi_0|<1)\), then \(\mathfrak R(A)\) is a proper negative subspace.

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
16 III 1961

REFERENCES

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^6 N. Bourbaki, Topological Vector Spaces, IL, 1959.
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