##
**The projection spectral theorem.**
*(English.
Russian original)*
Zbl 0577.47019

Russ. Math. Surv. 39, No. 4, 1-62 (1984); translation from Usp. Mat. Nauk 39, No. 4(238), 3-52 (1984).

This work is a continuation of several previous papers by the same author, all those devoted to the so-called “projective” spectral theorem. That is a natural variant of the spectral theorem for selfadjoint (or normal) operators on Hilbert spaces which is very useful for obtaining the generalized eigenvector expansions both in the discrete and continuous form. That variant is closely related to distribution theory and Fourier transform. The basic Hilbert space \(H_ 0\) is considered as intermediate in a chain of the following type
\[
H_-\supseteq H_ 0\supseteq H_+\supseteq D \tag{1}
\]
where \(H_+\) and \(H_-\) are Hilbert spaces (the space of basic and generalized vectors respectively), the inclusion \(H_+\subseteq H_ 0\) is a quasi-nuclear mapping (i.e. Hilbert-Schmidt) and \(D\) is a linear topological space, topologically imbedded (i.e. densely and continuous) in \(H_+\). A selfadjoint operator \(A\) is called standardly related to chain (1) if its domain \(D(A)\) contains \(D\), \(D(A)\supseteq D\), and the restriction \(A\upharpoonright D\) is continuous from \(D\) to \(H_+\). If \(A\) is standardly related to (1) then the resolution \(E\) of the identity of \(A\), considered as an operator measure from \(H_+\) to \(H_-\) may be differentiated with respect to a scalar measure \({\mathcal B}(R^ 1)\ni \alpha \to \rho (\alpha)\), that is \(dE/d\rho(\lambda)=P(\lambda):H_+\to H_-\) such that:
\[
1=\int_{R^ 1}P(\lambda)\,d\rho (\lambda),\quad A=\int_{R^ 1}\lambda \cdot P(\lambda)\,d\rho(\lambda). \tag{2}
\]
In such a situation for \(\rho\)-almost \(\lambda\in R^ 1,\) the range \({\mathcal R}(P(\lambda))\) consists of generalized eigenvectors \(\phi \in H_-\) corresponding to \(\lambda\) (i.e. \(A\phi =\lambda\phi\) in a generalized sense). Therefore \(P(\lambda)\) projects onto the corresponding generalized eigenspace. Sometimes it is more convenient to consider instead of (1) a ”nuclear” chain:
\[
\Phi '\supseteq H_ 0\supseteq \Phi\tag{3}
\]
where \(\Phi\) is a nuclear space topologically imbedded in \(H_ 0\) and \(\Phi\) ’ is the adjoint space of all conjugate linear continuous functionals on \(\Phi\). The spectral theorem has an analogous form in that context if \(D(A)\supseteq \Phi\) and \(A\upharpoonright \Phi\) is continuous.

The projective variant has a natural extension to finite families of commuting selfadjoint operators (i.e. having commuting resolutions of the identity). For families \(A=(A_ 1,...,A_ p)\), we must take \(\lambda =(\lambda_ 1,...,\lambda_ p)\) and \(R^ 1\) must be replaced by \(R^ p\).

The present work is devoted to obtaining the formulas analogous to (2) for the case of families \((A_ x)_{x\in X}\) of commuting selfadjoint operators of an arbitrary cardinal. This time there come difficulties especially when X is uncountable. The joint resolution of the identity must be a measure on \(R^ X\) with projection values and is constructed by following the Kolmogorov’s idea of extending a measure; however the extended \(\sigma\)-algebra of sets may be too poor as compared to the usual Borel \(\sigma\)-algebra \({\mathcal B}(R^ X)\) of the Tikhonov product \(R^ X\). Instead of a generalized eigenvector we have to consider the joint generalized eigenvectors \(\phi \in H_-\) such that \(A_ x\phi =\lambda(x)\phi\), \(\forall x\in X\) in a generalized sense. A good definition of the spectrum of A (the joint spectrum) is as the set of all generalized joint eigenvalues corresponding to all joint generalized eigenvectors of A.

The integral representation becomes \[ A_ x=\int_{sp A}\lambda (x)\,dE(\lambda(.)),\quad x\in X. \tag{4} \] The formula (4) has interesting consequences if \(x\to A_ x\) is an operator representation of an algebraic object: group, linear space; \(A_ x\) might also satisfy a differential or differential-functional relation with respect to x. The corresponding joint eigenvalues will then satisfy the corresponding scalar relations and the ”multiplicity” in (4) may be decreased. A classical prototype of such a situation is the Stone’s theorem and its generalizations.

The work is organized as follows. In section 1 the joint resolution of the identity for operators \(A_ x\) is constructed and studied. The construction reminds the standard constructions of measure theory but there are also unexpected facts: for instance two joint commuting resolutions of the identity cannot be always multiplied. In section 2 the differentiation of a joint resolution is studied as an operator measure from \(H_+\) to \(H_-\). In section 3 the projective spectral theorem is proved in the whole generality in a greatly improved exposition. The work is concluded with some interesting applications.

The projective variant has a natural extension to finite families of commuting selfadjoint operators (i.e. having commuting resolutions of the identity). For families \(A=(A_ 1,...,A_ p)\), we must take \(\lambda =(\lambda_ 1,...,\lambda_ p)\) and \(R^ 1\) must be replaced by \(R^ p\).

The present work is devoted to obtaining the formulas analogous to (2) for the case of families \((A_ x)_{x\in X}\) of commuting selfadjoint operators of an arbitrary cardinal. This time there come difficulties especially when X is uncountable. The joint resolution of the identity must be a measure on \(R^ X\) with projection values and is constructed by following the Kolmogorov’s idea of extending a measure; however the extended \(\sigma\)-algebra of sets may be too poor as compared to the usual Borel \(\sigma\)-algebra \({\mathcal B}(R^ X)\) of the Tikhonov product \(R^ X\). Instead of a generalized eigenvector we have to consider the joint generalized eigenvectors \(\phi \in H_-\) such that \(A_ x\phi =\lambda(x)\phi\), \(\forall x\in X\) in a generalized sense. A good definition of the spectrum of A (the joint spectrum) is as the set of all generalized joint eigenvalues corresponding to all joint generalized eigenvectors of A.

The integral representation becomes \[ A_ x=\int_{sp A}\lambda (x)\,dE(\lambda(.)),\quad x\in X. \tag{4} \] The formula (4) has interesting consequences if \(x\to A_ x\) is an operator representation of an algebraic object: group, linear space; \(A_ x\) might also satisfy a differential or differential-functional relation with respect to x. The corresponding joint eigenvalues will then satisfy the corresponding scalar relations and the ”multiplicity” in (4) may be decreased. A classical prototype of such a situation is the Stone’s theorem and its generalizations.

The work is organized as follows. In section 1 the joint resolution of the identity for operators \(A_ x\) is constructed and studied. The construction reminds the standard constructions of measure theory but there are also unexpected facts: for instance two joint commuting resolutions of the identity cannot be always multiplied. In section 2 the differentiation of a joint resolution is studied as an operator measure from \(H_+\) to \(H_-\). In section 3 the projective spectral theorem is proved in the whole generality in a greatly improved exposition. The work is concluded with some interesting applications.

Reviewer: Şt. Frunză

### MSC:

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |