Spectral synthesis

The reconstruction of the invariant subspaces of a family of linear operators from the eigen or root subspaces of this family contained in such subspaces. More precisely, let $ {\mathcal A} $ be a commutative family of operators on a topological vector space $ X $ and let $ \sigma _ {p} ( {\mathcal A} ) $ be its point spectrum, i.e. the set of numerical functions $ \lambda = \lambda ( A ) $ on $ {\mathcal A} $ for which the eigen subspaces

$$ N _ {\mathcal A} ( \lambda ) = \cap _ {A \in {\mathcal A} } \mathop{\rm Ker} ( A- \lambda ( A) I) $$

are distinct from zero, and let

$$ K _ {\mathcal A} ( \lambda ) = \cap _ {A \in {\mathcal A} } \cup _ {n \in \mathbf N } \mathop{\rm Ker} ( A- \lambda ( A) I) ^ {n} $$

be the root subspaces corresponding to the points $ \lambda \in \sigma _ {p} ( {\mathcal A} ) $( cf. Spectrum of an operator). A subspace $ L \subset X $ which is invariant under $ {\mathcal A} $ admits spectral synthesis if $ L $ coincides with the closure of the root subspaces contained in it. If all $ {\mathcal A} $- invariant subspaces admit spectral synthesis, then it is said that the family $ {\mathcal A} $ itself admits spectral synthesis.

Examples of families admitting spectral synthesis are as follows: any compact commutative group of operators on a Banach space and, more generally, any group with relatively compact trajectories. If $ \mathop{\rm dim} X < \infty $, then every one-element family admits spectral synthesis in view of the existence of the Jordan decomposition. In the general case, for an operator $ A $ to admit spectral synthesis it is necessary at least to require that the whole of $ X $ admits spectral synthesis with respect to $ A $, that is, $ A $ should have a complete system of root subspaces. But this condition is not sufficient, even for normal operators on a Hilbert space. In order that a normal operator $ A $ admits spectral synthesis it is necessary and sufficient that $ \sigma _ {p} ( A) $ does not contain the support of a measure orthogonal to the polynomials. This condition holds if and only if for any domain $ G \subset \mathbf C $ there is an analytic function $ f $ in $ G $ for which

$$ \sup _ {z \in G } | f( z) | < \sup _ {z \in G \cap \sigma _ {p} ( A) } | f( z) | . $$

In particular, unitary complete and self-adjoint complete operators (cf. Complete operator; Self-adjoint operator; Unitary operator) admit spectral synthesis. Spectral synthesis is also possible for complete operators that are "close" to unitary or self-adjoint ones (such as dissipative operators, cf. Dissipative operator, with a nuclear imaginary component, and operators with spectrum on a circle and with normal growth of the resolvent as one approaches the circle).

The completeness of the system of root subspaces does not guarantee spectral synthesis of invariant subspaces even if one imposes the further condition that the operator be compact: The restriction of a complete compact operator to an invariant subspace need not have eigenvectors and can even coincide with any compact operator given in advance.

The problems of spectral synthesis of invariant subspaces include not only the clarification of the possibility of approximating their elements by linear combinations of root vectors, but also the construction of an approximating sequence and the estimation of its rate of convergence. In the case of operators with a countable spectrum, the approximating sequence is usually constructed by averaging the sequence of partial sums of the formal Fourier series $ x \approx \sum _ {\lambda \in \sigma _ {p} ( A) } \epsilon _ \lambda x $, where $ \epsilon _ \lambda $ is the Riesz projector:

$$ \epsilon _ \lambda x = \frac{1}{2 \pi i } \int\limits _ {\Gamma _ \lambda } ( z- A) ^ {-} 1 dz. $$

Here, $ \Gamma _ \lambda $ is a contour separating the point $ \lambda \in \sigma _ {p} ( A) $ from the rest of the spectrum.

If a space $ X $ consists of functions on a locally compact Abelian group and $ {\mathcal A} $ coincides with the family of all shift operators, then the eigenspaces for $ {\mathcal A} $ are the one-dimensional subspaces generated by the characters of the group. Thus, the theory of spectral synthesis of invariant subspaces includes the classical problems of harmonic synthesis on a locally compact Abelian group (see Harmonic analysis, abstract), which consists of finding conditions under which the subspaces that are invariant under the translations in some topological vector space of functions on a group are generated by the characters contained in them. In particular, the possibility of spectral synthesis on compact groups or, more generally, in spaces of almost-periodic functions on groups is a consequence of the result stated above on the spectral synthesis for groups of operators with relatively compact trajectories. Moreover, the problems of spectral synthesis are closely connected with problems of synthesis of the ideals in a regular commutative Banach algebra: A closed ideal is the intersection of maximal ones ( "it admits spectral synthesis" ) if and only if its annihilator in the adjoint space admits spectral synthesis with respect to the family of operators adjoint to the operators of multiplication by elements of the algebra.

The above definition of spectral synthesis can be extended in such a way that that it also covers families of operators without an extensive point spectrum (and even non-commutative families). In that case it is replaced by the requirement of a one-to-one correspondence between the invariant subspaces and the spectral characteristics of the restrictions to these subspaces of a given family of operators. In this sense one talks of spectral synthesis for modules over a regular commutative Banach algebra, and for representations of a locally compact Abelian group.

References

[1]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)
[2]	N.K. Nikol'skii, "Invariant subspaces in the theory of operators and theory of functions" J. Soviet Math. , 5 : 2 (1976) pp. 129–249 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 199–412
[3]	J.J. Benedetto, "Spectral synthesis" , Teubner (1975)

Comments

According to [a2], p. 140, the term "spectral synthesis" was introduced around 1947 by A. Beurling. Since then it has been a subject of much research in commutative harmonic analysis, i.e. in the context of the commutative Banach algebra $ L _ {1} ( G) $, $ G $ a locally compact Abelian group. The elements of the dual group $ \widehat{G} $ can be identified with the closed maximal ideals of $ L _ {1} ( G) $. The cospectrum of a closed ideal $ I $ in $ L _ {1} ( G) $ is the closed set in $ \widehat{G} $ consisting of all closed maximal ideals containing $ I $. To every closed subset $ E $ of $ \widehat{G} $ corresponds a natural closed ideal in $ L _ {1} ( G) $ having $ E $ as cospectrum, namely the intersection of all closed maximal ideals corresponding to the points of $ E $. $ E $ is called a set of spectral synthesis (or a Wiener set, [a2]) if this intersection is the only closed ideal having $ E $ as cospectrum. The classical approximation theorem, proved for $ G = \mathbf R $ by N. Wiener (1932), can be stated as: The empty set is a set of spectral synthesis.

The first example of a set that is not a set of spectral synthesis (also called a "set of non spectral synthesisset of non spectral synthesis" ) was obtained in 1948 by L. Schwartz, who showed that spheres in $ \widehat{G} = \mathbf R ^ {n} $( $ n \geq 3 $) are such. That sets of non spectral synthesis exist in $ \widehat{G} $ for all non-compact $ G $ was proved by P. Malliavin (1959). A completely-different proof of this fact, using tensor algebra, was obtained in 1965 by N.Th. Varopoulos. A famous unsolved problem in this area is whether the union of two sets of spectral synthesis is again such a set (the union problem). See [1], [3], [a1], [a2] for many more details.

References

[a1]	C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5
[a2]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)