MATHEMATICS

V. E. LYANTSE

ON ONE GENERALIZED NOTION OF A SPECTRAL OPERATOR

(Presented by Academician A. N. Kolmogorov, 28 VIII 1961)

The spectral theory of non-self-adjoint operators for which the space decomposes into a direct (discrete or continuous) sum of invariant subspaces was developed with great completeness by N. Dunford and others \((^1)\). The decomposition of the space into a direct sum of invariant subspaces is equivalent to the unconditional convergence of expansions in eigenvectors and associated (generalized) vectors. However, verifying that the convergence is unconditional is, as a rule, very difficult. Therefore the existing criteria for the “spectrality” of an operator \((^1)\) are not very effective. Moreover, unconditional convergence does not always occur.

In the present paper an attempt is made to construct a spectral theory under the assumption only of completeness of the system of invariant subspaces. Our basic idea is very simple. We immerse the original space \(\mathfrak H\) into the projective limit \(\widehat{\mathfrak H}\) of the system of invariant subspaces \((\mathfrak H^\Delta)\). In the general case, expansions in eigenvectors diverge in the topology of the space \(\mathfrak H\), but they converge in the topology of the projective limit \(\widehat{\mathfrak H}\).

1. Generalized decompositions of the identity.

1.1. By a (generalized) decomposition of the identity of a Hilbert space \(\mathfrak H\) we shall mean an operator-valued set function \(P:\Delta \to P(\Delta)\) possessing the following properties: a) the function \(P\) is defined on a certain ring \(D(P)\) (which, generally speaking, is not a field or a \(\sigma\)-ring) of Borel sets \(\Delta\) of the complex plane \(\Lambda\), and the ring \(D(P)\) contains every Borel subset of each of its elements; b) the values of the function \(P\) are linear bounded operators \(P(\Delta)\) mapping the whole space \(\mathfrak H\) into itself, and: 1) \(P(\Delta_1)P(\Delta_2)=P(\Delta_1\cap\Delta_2)\) for any \(\Delta_1,\Delta_2\in D(P)\); 2) for any decomposition of a set \(\Delta\in D(P)\) into disjoint parts \(\Delta_1,\Delta_2,\ldots\in D(P)\) and for any \(x,y\in\mathfrak H\),
\[ \sum_k (P(\Delta_k)x,y)=(P(\Delta)x,y); \]
3) if \(P(\Delta)x=0\) for all \(\Delta\in D(P)\), then \(x=0\); 4) if \(P^*(\Delta)x=0\) for all \(\Delta\in D(P)\), then \(x=0\), \(P^*(\Delta)=[P(\Delta)]^*\).

It is not difficult to construct an example of a generalized decomposition of the identity \(P\) that is not equivalent to a decomposition of the identity of a normal operator, i.e. one for which there exists no such continuous automorphism \(M\) of the space \(\mathfrak H\) that all the projectors \(M^{-1}P(\Delta)M\), \(\Delta\in D(P)\), are self-adjoint operators.

1.2. For an arbitrary set \(\Delta\subset\Lambda\) put
\[ K_\Delta=\sup\{\|P(\delta)\|:\delta\subset\Delta,\ \delta\in D(P)\}. \]
If \(\Delta\in D(P)\), then \(K_\Delta<\infty\).

1.3. Denote by \(D_0(P)\) the class of all Borel sets \(\Delta\subset\Lambda\) for which \(K_\Delta<\infty\). For every \(\Delta\in D_0(P)\) the sequence of operators \((P(\delta))\) converges in the strong sense when the set \(\delta\) runs through the upward-directed, with respect to inclusion, system of subsets of the set \(\Delta\) belonging to the class \(D(P)\). Put
\[ P(\Delta)=\lim\{P(\delta):\delta\uparrow\Delta,\ \delta\in D(P)\}. \]
The extended function \(P\) is also a decomposition of the identity, and the extension constructed above is maximal (in the sense that \(P\) still remains a decomposition of the identity).

1.4. A class of sets \(D \subset D(P)\) shall be called admissible if the restriction of the function \(P\) to the class \(D\) is also a resolution of the identity. The intersection of any two admissible classes is an admissible class.

1.5. If \(D\) is an admissible class, then the manifold
\[ \bigcup_{\Delta \in D} P(\Delta)\mathfrak H \]
is dense in \(\mathfrak H\).

1.6. If \(P\) is a resolution of the identity defined on the class \(D(P)\), then the function
\[ P^*: \Delta \to P^*(\Delta)=[P(\Delta)]^*, \]
defined on the same class of sets \(D(P)\), is also a resolution of the identity.

2. Basic and generalized elements

2.1. Let \(P\) be a resolution of the identity of the space \(\mathfrak H\); \(D(P)\) the domain of definition of the function \(P\) (not necessarily maximal; \(D(P)\subset D_0(P)\), see 1.3). For each \(\Delta\in D(P)\) put
\[ \mathfrak H^\Delta=P(\Delta)\mathfrak H. \]
The (generalized) sequence \((\mathfrak H^\Delta)_{\Delta\in D(P)}\) of Hilbert spaces \(\mathfrak H^\Delta\subset\mathfrak H\) increases with respect to \(\Delta\) in the sense that
\[ P(\Delta_1)\mathfrak H^{\Delta_1}\subset \mathfrak H^{\Delta_2},\quad \text{if } \Delta_1\subset \Delta_2. \]
By the space of basic elements \(\widetilde{\mathfrak H}\) of the Hilbert space \(\mathfrak H\), corresponding to \(P\) and \(D(P)\), we shall mean the inductive limit
\[ \widetilde{\mathfrak H}=\lim \operatorname{ind}_{\Delta}(\mathfrak H^\Delta,\,P(\Delta)). \]
The space \(\widetilde{\mathfrak H}\) consists of the elements of the union
\[ \bigcup_{\Delta\in D(P)} \mathfrak H^\Delta, \]
linear operations in \(\widetilde{\mathfrak H}\) are induced from \(\mathfrak H^\Delta\), \(\Delta\in D(P)\), and the defining system of neighborhoods of zero in \(\widetilde{\mathfrak H}\) is formed by sets of the form
\[ \widetilde U=\bigcup_{\Delta\in D(P)} U^\Delta, \]
where \(U^\Delta\) is a neighborhood of zero in \(\mathfrak H^\Delta\).

2.2. Note that the (generalized) sequence \((\mathfrak H^\Delta)_{\Delta\in D(P)}\) of Hilbert spaces \(\mathfrak H^\Delta\subset\mathfrak H\) decreases with respect to \(\Delta\) in the sense that
\[ P(\Delta_1)\mathfrak H^{\Delta_2}\subset \mathfrak H^{\Delta_1}, \quad \text{if } \Delta_1\subset \Delta_2. \]

By the space of generalized elements \(\widehat{\mathfrak H}\) of the Hilbert space \(\mathfrak H\), corresponding to \(P\) and \(D(P)\), we shall mean the projective limit
\[ \widehat{\mathfrak H}=\lim \operatorname{pr}_{\Delta}(\mathfrak H^\Delta;\,P(\Delta)). \]
The space \(\widehat{\mathfrak H}\) consists of (generalized) sequences
\[ \hat x=(x_\Delta)_{\Delta\in D(P)} \]
such that \(x_\Delta\in \mathfrak H^\Delta\) and
\[ P(\Delta_1)x_{\Delta_2}=x_{\Delta_1},\quad \text{if } \Delta_1\subset \Delta_2. \]
By definition
\[ (x_\Delta)+(y_\Delta)=(x_\Delta+y_\Delta) \]
and
\[ \alpha(x_\Delta)=(\alpha x_\Delta),\quad \alpha\in\Lambda. \]
The defining system of neighborhoods of zero in \(\widehat{\mathfrak H}\) is formed by sets of the form
\[ \widehat U=\{\hat x:\hat x\in\widehat{\mathfrak H},\,P(\Delta_0)\hat x\in U^{\Delta_0}\}, \]
where \(U^{\Delta_0}\) is a neighborhood of zero in \(\mathfrak H^{\Delta_0}\) (depending on \(\widehat U\)), and
\[ P(\Delta_0)(x_\Delta)=x_{\Delta_0};\quad \Delta,\Delta_0\in D(P). \]

2.3. Let \(\widetilde{\mathfrak H}_*\) be the space of basic elements of the space \(\mathfrak H\), corresponding to the resolution of the identity \(P^*\), defined on the class \(D(P^*)=D(P)\) (see 1.6). The space of generalized elements \(\widehat{\mathfrak H}\) is the space of linear continuous functionals on \(\widetilde{\mathfrak H}_*\). Namely, the formula
\[ \langle \tilde x_*,\hat x\rangle=(\tilde x_*,P(\Delta)\hat x) \]
for all \(\tilde x_*\in\widetilde{\mathfrak H}_*\), \(\Delta\in D(P)\), establishes a one-to-one correspondence between functionals \(\tilde x_*\in\widetilde{\mathfrak H}_*'\) and generalized elements \(\hat x\in\widehat{\mathfrak H}\) (recall that \(P(\Delta_0)(x_\Delta)=x_{\Delta_0}\); \(\langle\, ,\,\rangle\) denotes the duality of the spaces \(\widetilde{\mathfrak H}_*\) and \(\widetilde{\mathfrak H}_*'\), and \((\, ,\,)\) is the scalar product in \(\mathfrak H\)).

2.4. For each \(x\in\mathfrak H\) there exists a unique element \(\hat x\in\widehat{\mathfrak H}\) such that
\[ P(\Delta)\hat x=P(\Delta)x \]
for all \(\Delta\in D(P)\). In this sense \(\mathfrak H\subset\widehat{\mathfrak H}\), and since, moreover, \(\widetilde{\mathfrak H}\subset\mathfrak H\), we have
\[ \widetilde{\mathfrak H}\subset\mathfrak H\subset\widehat{\mathfrak H}. \]
The manifold \(\widetilde{\mathfrak H}\) is dense in the topological space \(\widehat{\mathfrak H}\).

2.5. For each \(\Delta\in D(P)\), by \(\widetilde P(\Delta)\) \((\widehat P(\Delta))\) we denote the restriction (extension) of the operator \(P(\Delta)\) to the manifold \(\widetilde{\mathfrak H}\) (\(\widehat{\mathfrak H}\): \(\widehat P(\Delta)\hat x=JP(\Delta)x\), where \(J\) is the operator of embedding of \(\mathfrak H\) into \(\widehat{\mathfrak H}\); this extension is at the same time an extension by continuity). For every \(x\in\widetilde{\mathfrak H}\) \((\hat x\in\widehat{\mathfrak H})\) we have
\[ \tilde x=\int \widetilde P(d\lambda)\tilde x =\lim_{\Delta\nearrow}\widetilde P(\Delta)\tilde x \quad \left( \hat x=\int \widehat P(d\lambda)\hat x =\lim_{\Delta\nearrow}\widehat P(\Delta)\hat x \right) \]
(\(\Delta\nearrow\) means that \(\Delta\) runs through the upward-directed, with respect to inclusion \(\subset\), system of mno-

\(D(P))\). Consequently, for every \(x\in\mathfrak H\) we have the expansion \(x=\int P(d\lambda)x\), where the integral exists in the sense of the topology of the projective limit \(\mathfrak H\).

3. Operators permutable with the resolution of the identity

3.1. A closed linear operator \(A\) with domain of definition \(\mathfrak D(A)\), dense in the space \(\mathfrak H\), will be called permutable with the resolution of the identity \(P\) if: \(\alpha)\) \(AP(\Delta)\supset P(\Delta)A\) for every \(\Delta\in D(P)\); \(\beta)\) the class \(D_A\) of all those sets \(\Delta\in D(P)\) for which \(P(\Delta)\mathfrak H\subset \mathfrak D(A)\) is an admissible class (see 1.4). If the operator \(A\) is permutable with the resolution of the identity \(P\), then, as is not difficult to see, the operator \(AP(\Delta)\) is bounded for every \(\Delta\in D(P)\). The totality of all operators permutable with \(P\) will be denoted by \(\mathfrak A(P)\).

3.2. Let \(\Delta\to \widetilde A(\Delta)\) be an operator-valued set function defined on some admissible class of sets \(D\), and suppose that for each \(\Delta\in D\), \(\widetilde A(\Delta)\) is a bounded linear operator \(\mathfrak H=\mathfrak D(A(\Delta))\to\mathfrak H\). In order that there exist an operator \(A\in\mathfrak A(P)\) for which \(AP(\Delta)=\widetilde A(\Delta)\), \(\Delta\in D\), it is necessary and sufficient that the condition

\[ P(\Delta_1)\widetilde A(\Delta_2)=\widetilde A(\Delta_2)P(\Delta_1)=A(\Delta_1\cap\Delta_2) \quad \text{for all } \Delta_1,\Delta_2\in D \tag{a} \]

be satisfied.

3.3. Let \(\Delta\to \widetilde A_i(\Delta)\) be functions satisfying the conditions of 3.2, and let \(A_i\in\mathfrak A(P)\) be the operator determined by this function, \(A_iP(\Delta)=\widetilde A_i(\Delta)\), \(i=1,2\). If \(\widetilde A_1(\Delta)=\widetilde A_2(\Delta)\) for all \(\Delta\in D_1\cap D_2\), then \(A_1=A_2\); in particular \(\mathfrak D(A_1)=\mathfrak D(A_2)\).

3.4. If \(A\in\mathfrak A(P)\), then \(A^*\in\mathfrak A(P^*)\) and \(D_A=D_{A^*}\).

3.5. Denote by \(\widetilde{\mathfrak A}(P)\) \((\widehat{\mathfrak A}(P))\) the class of all linear continuous operators \(\mathfrak H\to\widetilde{\mathfrak H}\) \((\widetilde{\mathfrak H}\to\mathfrak H)\) that are permutable with each of the operators \(\widetilde P(\Delta)\) \((\widehat P(\Delta))\), \(\Delta\in D(P)\). In the following propositions a description is given of the relations existing among the elements of the classes \(\widetilde{\mathfrak A}(P)\), \(\mathfrak A(P)\), and \(\widehat{\mathfrak A}(P)\).

\(1^\circ.\) Let \(\widetilde A\in\widetilde{\mathfrak A}(P)\). The operator \(\widetilde A\), considered in the space \(\mathfrak H\supset\widetilde{\mathfrak H}\), admits a closure \(A\); moreover \(A\in\mathfrak A(P)\).

\(2^\circ.\) Let \(A\in\mathfrak A(P)\). Since the restriction of the function \(P\) to the class \(D_A\) (see 3.1\(\beta\)) is also a resolution of the identity, we shall assume that \(D(P)=D_A\). Under this assumption \(\widetilde{\mathfrak H}\subset\mathfrak D(A)\). If \(\widetilde A\) is the restriction of the operator \(A\) to \(\widetilde{\mathfrak H}\), then \(\widetilde A\in\widetilde{\mathfrak A}(P)\).

\(3^\circ.\) Let \(\widetilde A\in\widetilde{\mathfrak A}(P)\). The operator \(\widetilde A\) is continuous in the sense of the topology induced in \(\widetilde{\mathfrak H}\) by the topology of the space \(\mathfrak H\supset\widetilde{\mathfrak H}\), and since \(\widetilde{\mathfrak H}\) is dense in \(\widehat{\mathfrak H}\), \(\widetilde A\) extends by continuity to the space \(\widehat{\mathfrak H}\). Denoting this extension by \(\widehat A\), we have \(\widehat A\in\widehat{\mathfrak A}(P)\).

\(4^\circ.\) Let \(\widehat A\in\widehat{\mathfrak A}(P)\). Put
\[ \mathfrak D(A)=\{\hat x:\hat x\in\mathfrak H,\ \widehat A\hat x\in\mathfrak H\},\qquad A\subset\widehat A. \]
Then \(A\in\mathfrak A(P)\).

3.6. The set \(\mathfrak A(P)\) is transformed into an algebra over the field of complex numbers if the algebraic operations on the operators \(A\in\mathfrak A(P)\) are defined by means of the corresponding operations on the (bounded) operators \(\widetilde A(\Delta)=AP(\Delta)\) (see 3.2 and condition (a)). The sets \(\widetilde{\mathfrak A}(P)\) and \(\widehat{\mathfrak A}(P)\) consist of operators defined on the whole space (\(\widetilde{\mathfrak H}\) or \(\widehat{\mathfrak H}\)), and therefore are algebras with respect to the usual operations on operators. The one-to-one correspondences \(A\to\widetilde A\) and \(A\to\widehat A\) (see 3.5) are isomorphisms of the algebras \(\mathfrak A(P)\), \(\widetilde{\mathfrak A}(P)\), and \(\widehat{\mathfrak A}(P)\). These correspondences are also continuous mappings if convergence of a sequence \((A_\alpha)\in\mathfrak A(P)\) is defined as convergence (in norm in \(\mathfrak H\)) of the sequences \((A_\alpha P(\Delta))\) for every \(\Delta\in D(P)\).

3.7. Proposition 3.2 can naturally be used for constructing analytic functions of elements of the algebra \(\mathfrak A(P)\). For every \(\Delta\in D(P)\)

we first construct the operator \(f(AP(\Delta))\), where \(f\) is a function analytic in some neighborhood of the spectrum of the operator \(A \in \mathfrak{A}(P)\); we verify that the function \(\Delta \to f(A(\Delta))\) satisfies condition (3.1), and therefore defines an operator \(f(A) \in \mathfrak{A}(P)\).

4. Spectral operators. A linear operator \(A: \mathfrak{D}(A)\to\mathfrak{H}\), \(\overline{\mathfrak{D}(A)}=\mathfrak{H}\), will be called spectral if there exists a resolution of the identity \(P\) such that \(A \in \mathfrak{A}(P)\) and, for every \(\Delta \in D_A\), the spectrum of the operator \(A\), considered on the invariant subspace \(P(\Delta)\mathfrak{H}\), is contained in the closure \(\overline{\Delta}\) of the set \(\Delta\). An operator \(A \in \mathfrak{A}(P)\) is spectral if and only if
\[ A=\int \lambda P(d\lambda)+N, \]
where the integral converges in the topology of \(\mathfrak{A}(P)\), while \(N\in\mathfrak{A}(P)\) and \(NP(\Delta)\) is a quasinilpotent operator for every \(\Delta\in D_A\).

Lviv Polytechnic Institute

Received
1 VII 1961

REFERENCES

1. N. Dunford, Sborn. per. Matematika, 4, 1, 1960.