Спектральный синтез на нульмерных локально компактных абелевых группах

Аннотация

Пусть G - нульмерная локально компактная абелева группа, все элементы которой компактны, C( G) - пространство всех непрерывных комплекснозначных функций на группе G . Замкнутое линейное подпространство H ⊆ C( G) называется инвариантным подпространством, если оно инвариантно относительно сдвигов τ y : f( x) ↦ f( x + y) , y ∈ G . В работе доказывается, что любое инвариантное подпространство H допускает спектральный синтез, то есть H совпадает с замыканием линейной оболочки всех содержащихся в H характеров группы G .

Полный текст

1. General definitions Let G be a locally compact Abelian group (LCA-group), F be a locally convex topological vector space that consists of complex-valued functions on the group G. This space is called a translation invariant space if it is invariant under translations (shifts) y : f(x) 7! f(x + y); f 2 F; y 2 G; and all operators y on the space F are continuous. A closed linear subspace H F is called an invariant subspace if y(H) H for any y 2 G. A continuous homomorphism of G into the multiplicative group C = C n f0g of nonzero complex numbers is called an exponential functions or generalized character on G. A continuous homomorphism of G into the group T := fz 2 C : jzj = 1g is called a character of G. Continuous homomorphisms of G into the additive group of comlex numbers are called additive functions. A function x 7! P(a1(x); : : : ; am(x)) on G is called a polynomial if P is a complex polynomial in m variables and a1; : : : ; am are additive functions. A product of a polynomial and an exponential function is called an exponential monomial, and linear combinations of of exponential monomials are called exponential polynomials. Let F be a translation invariant space on G and H be an invariant subspace in F . D e f i n i t i o n 1.1. An invariant subspace H admits spectral synthesis if H coincides with the closed linear span in F of all exponential monomials that belong to H. We say that a translation invariant space F has the spectral synthesis property if any invariant subspace H F admits spectral synthesis. 2. Examples of spectral synthesis In this section we give some examples of spectral synthesis. 1. G = (R; +) Any exponential monomial on R has the form f(x) = P(x) e x , where x 2 R, 2 C, P(x) is a polynomial. The function spaces C(R) of all continuous functions and E(R) = C1(R) of all infinitely differentiable functions (all classical function spaces are equipped with their usual topologies) have the spectral synthesis property. This is result of L. Schwartz [1]. Some other examples of functions spaces on R with spectral synthesis property were studied in the papers of J. E. Gilbert [2] and S. S. Platonov [3]. 2. G = (Rn; +) , n 2 Any exponential monomial on Rn has the form f(x)=P(x) e x , where x=(x1; : : : ; xn)2 Rn , = ( 1; : : : ; n) 2 Cn; x = 1x1 + + nxn , P(x) is a polynomial in x . In [1] L. Schwartz conjectured that the spaces C(Rn) and E(Rn) = C1(Rn) have the spectral synthesis property. This conjecture turned out to be false. In 1975, D. I. Gurevich [4] costructed an example of an invariant subspace H E(R2) containing no exponential monomials. Nevertheless, L. Schwartz [5] proved that the space S0(Rn) of all tempered distributions on Rn has the spectral synthesis property. 3. G is a discrete group For the case when G is a discrete group, the most natural function space is the space C(G) consisting of all complex-valued functions on G with the topology of pointwise 452 S. S. Platonov convergence. The case G = Zn was studied by M. Lefranc [6]. He proved that the space C(Zn) has the spectral synthesis property. Some results about the spectral synthesis on the discrete groups were considered in [7]. In particular, the space C(G) has the spectral synthesis property if G is a finitely generated Abelian group [8] or a torsion Abelian group [9]. In [10] M. Laczkovich and L. Sz´ekelyhidi proved that the spectral synthesis in the space C(G) holds on a discrete Abelian group G if and only if the torsion free rank of G is finite. For the case when G is a finitely generated discrete Abelian group and F is the space of all exponential growth functions on G the spectral synthesis property was proved in [11]. 3. Main results Let G be a LCA-group. An element x 2 G is called a compact element if the smallest closed subgroup of G, which contains x , is compact. Let G be a LCA-group, such that all elements of G are compact. Any generalized character of G is a usual character and any additive function on G is zero. Any exponential monomial on G has the form (x) , where 2 C, (x) is a character of G. P r o p o s i t i o n 3.1. Let F be a translation invariant space on G, H be an invariant subspace in F . If G is a LCA-group, such that all elements of G are compact, then H admits spectral synthesis if and only if H coicides with the closed linear span in F of all characters of G that belong to H. For any LCA-group G let bG be the set of all characters of G. The set bG is a LCAgroup (dual group of G) with compact-open topology and multiplication being defined as the pointwise multiplication of functions. For any invariant subspace H F , the set (H) := f 2 bG : 2 Hg: is called the spectrum of H. If G is a LCA-group, such that all elements of G are compact, and invariant subspace H admits spectral synthesis, then H can be recovered uniquely by its spectrum (H) . A locally compact topological space X is called zero-dimensional if compact open subsets of X form a basis of topology. A locally compact Hausdorff topological space X is zerodimensional if and only if X is totally disconnected, that is any subset of X , which contains more then one point, is disconnected. Theorem 3.1. Let G be a locally compact zero-dimensional Abelian group, such that all elements of G are compact. Then: 1) the space C(G) of all continuous functions on G has the spectral synthesis property; 2) a subset bG is the spectrum of some invariant subspace of C(G) if and only if is closed subset of bG . 4. Some examples of zero-dimensional LCA-groups, all elements of which are compact 1. Let fnkgk2Z be a two-side sequence, nk 2 N, nk > 2 . Let eG = M k2Z Znk ; SPECTRAL SYNTHESIS 453 where Zn is the cyclic group of order n . Every Znk is a discrete group and eG is a compact group. Any element of eG has the form x = fxkgk2Z; xk 2 Znk : Let G be a subgroup of eG that consist of all elements x = fxkg 2 eG : 9N(x) 2 Z 8k < N(x) xk = 0: The group G is locally compact, zero-dimensional and all elements of G are compact. If nk = 2 8k 2 Z, then we have the locally compact Cantor dyadic group. The harmonic analysis on this group closely connected with Fourier-Walch harmonic analysis (see [12]). 2. Let Qp be the group of p -adic numbers. Any element x 2 Qp can be identified with a formal series x = X k>N(x) xkpk; xk 2 f0; 1; : : : ; p 1g; N(x) 2 Z: The group Qp is locally compact, zero-dimensional and all elements of G are compact. Also, for any two-side sequence a = (ak)k2Z , ak 2 N, ak > 2 , there exist the group Qa of generalized a -adic numbers (see [13]). The group Qa is locally compact, zero-dimensional and all elements of G are compact. A zero-dimensional LCA-group G with countable base of topology, such that all elements of G are compact, is called a Vilenkin group. Harmonic analysis on such groups was studied in [14]. 5. On the ideal structure of algebras of locally constant functions Let X be a zero-dimensional Hausdorff locally compact topological space. Let co(X) be the set of all compact open subsets of X . The set co(X) forms a basis of topology of X . Any finite set = fU1; : : : ;Ung of mutually disjoint subsets Ui 2 co(X) is called a discrete system of subsets of X . Let M(X) be the set of all discrete systems of subsets of X . For = fU1; : : : ;Ung 2 M(X) , the support of is the set supp := [n i=1 Ui: A function f on X is called locally constant if for any x 2 X there exist neighbourhood U = U(x) of x on which f is constant. Denote by D(X) the set of all locally constant complex-valued functions on X with compact support. The set D(X) is a linear space. Now we define a topology on D(X) . For any 2 fU1; : : : ;Ung 2 M(X) let D (X) be the set of functions of the form f = Pn i=1 ci IUi ; where ci 2 C, IU is the characteristic function of U . The set D (X) is n -dimensional vector space. With respect to the uniform norm kfk1 := sup x2X jf(x)j the set D (X) is a Banach space. We equip the space D(X) = [ 2M(X) D (X) 454 S. S. Platonov with the topology of inductive limits of the Banach spaces D (X) , that is a topology of D(X) is the weakest locally convex topology for which all inclusions D (X) D(X) are continuous. Then D(X) is locally convex space.With respect to the pointwise multiplication of functions, D(X) is a topological algebra. Let I be an ideal of the algebra D(X) . Denote by N(I) the set of zeros of all functions from I , that is N(I) := fx 2 X : f(x) = 0 8x 2 Ig: The set N(I) is called zero set of I . For any closed subset A X denote by IA the set of all functions f 2 D(X) , such that f(x) = 0 for any x 2 A. The set IA is a closed ideal of D(X) . Theorem 5.1. Let I be an ideal of the algebra D(X) then IN(I) = I: Corollary 5.1. Any ideal of the topological algebra D(X) is closed. 6. The proof of Theorem 3.1 Let G be a zero-dimensional LCA-group, C(G) be the set of all continuous functions on G, D(G) be the set of locally constant functions with compact support on G. By Mc(G) we denote the set of complex-valued Radon measures with compact support on G. The space Mc(G) can be identified with the dual space of C(G) with respect to the duality h ; fi := Z G f(x) d (x); f 2 C(G); 2Mc(G): The space Mc(G) is a locally convex space with respect to the weak topology (Mc(G);C(G)) . Let 1; 2 2Mc(G) . A convolution 1 2 is defined by formula h 1 2; 'i := Z G Z G '(x + y) d 1(x) d 2(y); where ' 2 C(G) . The set Mc(G) is a commutative topological algebra with convolution as multiplication. For any closed linear subspace H C(G) , let H? be its annihilator in Mc(G) that is H? := f 2Mc(G) : h ; fi = 0 8f 2 Hg: The mapping H 7! H? is one-to-one correspondence between the set of all invariant subspaces of C(G) and the set of all closed ideals of topological algebra Mc(G) . Let D(G) be the set of all locally constant complex-valued functions on G with compact support. The set D(G) is a commutative topological algebra with convolution as multiplication: (f1 f2)(x) = Z G f1(x y)f2(y)dy: f1; f2 2 D(G): We will denote this topological algebra by Dconv(G) . SPECTRAL SYNTHESIS 455 For any topological algebra A we will denote by s(A) the set of all closed ideals of A. In particular we have the sets s(Mc(G)) and s(Dconv(G)) . Using identification a function f 2 D(G) with the measure f(x) dx , we have inclusion D(G) Mc(G) . The maps : s(Mc(G)) 7! s(Dconv(G)) and ~ : s(Dconv(G)) 7! s(Mc(G)) are defined by formulas: (H) := H \\ D(G); H 2 s(Mc(G)); ~ (H0) := [H0]; H0 2 s(Dconv(G)); where [H0] is the closure of H0 in the space Mc(G) . P r o p o s i t i o n 6.1. The mapping is a biection of set s(Mc(G)) onto the set s(Dconv(G)) . The inverse mapping 1 coincide with ~ . Let G be a LCA-group and bG be the dual group. It can be proved that LCA-group G is zero-dimensional group, all elements of which are compact, if and only if the dual group bG is zero-dimensional group, all elements of which are compact. The Fourier transform of a function f 2 L1(G) is the function b f on the dual group bG which is defined by formula b f( ) := Z G f(x) (x) dx; 2 b G: In particular, the Fourier transform is defined for any function f 2 D(G) . The mapping : f 7! b f is also called the Fourier transform. P r o p o s i t i o n 6.2. If G is a is zero-dimensional group, all elements of which are compact, then the Fourier transform is an isomorphism of the topological vector space D(G) into the topological vector space D(bG ) . Corollary 6.1. The mapping is an isomorphism of topological algebra Dconv(G) into the topological algebra Dmult(bG ) . P r o o f of Theorem 3.1 Let H be an invariant subspace of C(G) , H? be its annihilator in Mc(G) , I = H? \\ D(G) , bI = (I) . Then I is a closed ideal of Dconv(G) , and bI is a closed ideal of Dmult(bG ) .We will say that the ideal bI corresponds to the invariant subspace H. Let 2 bG . One can prove that 2 H if and only if the point belongs to zero set of the ideal bI . Thus the spectrum (H) of invariant subspace H is the same as zero set N(bI) of corresponding ideal bI Dmult(bG ) . Let H be an invariant subspace of C(G) . Denote by H1 a closed linear subspace of C(G) , that coincides with the closed linear span in C(G) of all characters of G that belong to H. Then H1 is also an invariant subspace of C(G) and (H) = (H1) . Let I1 = H? 1 \\ D(G) , bI1 = (I1) . Since N(bI) = N(bI1) then we have bI = bI1 by Theorem 5.1, and from Proposition 6.1 we have H = H1 . This completes the proof of Theorem 3.1.