# Generalized displacement operators

generalized shift operators, hypergroup

A concept arising from the axiomatization of certain properties of displacement operators on spaces of functions on a group. Such important mathematical concepts as the convolution of functions, a group algebra, a positive-definite function, an almost-periodic function, etc., can be formulated in terms of group-displacement operators. In the framework of the theory of generalized displacement operators far-reaching generalizations can be obtained of the fundamental principles and results concerning the concepts listed above. In particular, the theory of generalized displacement operators has substantial applications in abstract harmonic analysis (cf. Harmonic analysis, abstract).

The terms "generalized displacement operator" and "hypergroup" are due to J. Delsarte (see [1][3]). Important ideas and a number of original results in this field are also due to him. The systematic construction of the theory of generalized displacement operators was mainly given in the work of B.M. Levitan (see, for example, [4][7]).

## Basic concepts.

Let $H$ be an arbitrary set and let $\Phi$ be some vector space of complex-valued functions defined on $H$. Suppose that to each element $x \in H$ there corresponds a linear operator $R ^ {x}$ on $\Phi$ such that for any fixed $h \in H$ the function $\psi ( x) = R ^ {x} \phi ( h)$ is in $\Phi$ for all $\phi \in \Phi$. The linear operator $\phi ( x) \mapsto \psi ( x) = R ^ {x} \phi ( h)$ on $\Phi$ is denoted by $L ^ {h}$( that is, $L ^ {h} \phi ( x) = R ^ {x} \phi ( h)$). The linear operators $R ^ {x}$ are called generalized displacement operators if the following conditions (axioms) are satisfied: 1) $R ^ {x} L ^ {y} = L ^ {y} R ^ {x}$ for any $x , y \in H$( associativity axiom); and 2) there exists a neutral element $e$ in $H$ such that $R ^ {e} = I$, where $I$ is the identity operator. In this case the set $H$ is called a hypergroup, so that the concepts of "a set of generalized displacement operators" and a "hypergroup" are equivalent. The operators $R ^ {x}$ are often called right displacement operators, while the $L ^ {x}$ are called left displacement operators.

Generalized displacement operators arise in an obvious way on any vector space of functions on an arbitrary semi-group with an identity that is invariant with respect to displacements, and in particular on a group. Let $R ^ {x} : \phi ( h) \mapsto \phi ( h \cdot x )$, where $h \cdot x$ is the product of $h$ and $x$ in the semi-group, and let $L ^ {x} : \phi ( h) \mapsto \phi ( x \cdot h )$; then the associativity axiom reduces to the associativity of multiplication in the semi-group, and the neutral element is the identity in the semi-group, so that the operators $R ^ {x}$ form a family of generalized displacement operators. Non-trivial examples are given below.

In the general case, the $L ^ {x}$ do not form generalized displacement operators, since the operator $L ^ {e}$ need not be the identity. However, $L ^ {e}$ is a projector, and its range $\widetilde \Phi$ is called the fundamental subspace in $\Phi$. The operators $L ^ {x}$ form a family of generalized displacement operators in $\widetilde \Phi$, and the symmetry between left and right displacement operators is restored. The second of the axioms for generalized displacement operators is often strengthened by requiring that $L ^ {e} = I$, that is $\widetilde \Phi = \Phi$. Conditions 1) and 2) are the most general axioms for generalized displacement operators. More restricted classes of generalized displacement operators can be selected by imposing additional conditions. If $R ^ {x} R ^ {y} = R ^ {y} R ^ {x}$ for all $x , y \in H$, then the $R ^ {x}$ are called commutative, in this case the hypergroup $H$ is also called commutative. If further assumptions are made about $H$, then new conditions arise in a natural way for the generalized displacement operators. For example, if $H$ is a locally compact space with a measure $m$, then it is usually required that the operators $R ^ {x}$ and $L ^ {x}$ act compatibly on the space $C ( H)$ of continuous functions on $H$, and on the spaces $L _ {p} ( H , m )$, $p \geq 1$, so that additional conditions of continuity type are imposed on $R ^ {x}$ and $L ^ {x}$; if $H$ is a smooth manifold, then conditions of differentiability type are imposed, etc. Different versions of the axioms are given in [1], [3]–, [8], [15][20].

## Examples of generalized displacement operators related to groups.

Delsarte generalized displacement operators. Let $G$ be a topological group, let $K$ be a compact group of automorphisms of $G$, and let $dk$ be the Haar measure on $K$ with $\int dk = 1$. In the space $\Phi = C ( G)$ of continuous functions on $G$, the generalized displacement operators $R ^ {x}$ are defined by the equation

$$R ^ {x} \phi ( g) = \int\limits _ { K } \phi ( g \cdot k ( x) ) dk ,$$

where $\phi \in \Phi$, $k ( x)$ is the image of $x \in G$ under the automorphism $k \in K$, and $g \cdot k ( x)$ is the product of the elements $g$ and $k( x)$ in $G$. The identity element of the group is the neutral element. Both axioms for generalized displacement operators are satisfied; if $G$ is commutative, then Delsarte generalized displacement operators are also commutative. The fundamental subspace $\widetilde \Phi$ consists of all functions constant on the orbits relative to the action of $K$, and the operators $R ^ {x}$ and $R ^ {y}$ coincide on $\Phi$ if $x$ and $y$ lie in the same orbit. Hence the orbit space $H$ can also be given the structure of a hypergroup, by identifying $\Phi$ with the space of continuous functions on $H$ and putting $R ^ {h} = R ^ {x}$, where $x$ is any element in the orbit $h$. If $G$ is $\mathbf R$, and the group of automorphisms consists of two elements (reflection in the origin and the identity mapping), then $R ^ {x} \phi ( t) = [ \phi ( t+ x ) + \phi ( t- x ) ] / 2$. In this case the fundamental subspace consists of the even functions, and the orbit space is identified with the semi-axis $0 \leq t < \infty$. Another special case of Delsarte generalized displacement operators is obtained when $G = K$ and $k ( x) = k ^ {-} 1 xk$; here the fundamental subspace consists of the central functions on $K$, and the hypergroup generated by the orbits, that is, by the conjugacy classes, is commutative.

### Double cosets with respect to a compact subgroup.

Let $G$ be a locally compact group, let $K$ be a compact subgroup, and let $H$ be the space of double cosets of $K$( with $g \in G$, such a coset contains all elements of the form $k _ {1} gk _ {2}$, where $k _ {1} , k _ {2} \in K$). If $K$ is a normal subgroup of $G$, then $H$ coincides with the quotient group $G / K$. Let $\Phi$ be the space of all continuous functions on $G$ for which $\phi ( k _ {1} gk _ {2} ) = \phi ( g)$ for any $k _ {1} , k _ {2} \in K$, $g \in G$. A generalized displacement operator is defined by the formula

$$R ^ {x} \phi ( g) = \int\limits _ { K } \phi ( xkg ) dk .$$

The space $\Phi$ can be identified with the space $C( H)$ of continuous functions on $H$, and, as for Delsarte generalized displacement operators, $H$ can be given the structure of a hypergroup. If $G$ is a linear semi-simple Lie group and $K$ is its maximal compact subgroup, then the hypergroup $H$ is commutative and is closely connected with the spherical functions on $G$( in particular, all spherical functions lie in $\Phi$).

In the examples described above, other function spaces may be considered instead of that of the continuous functions (see [8], [13], [15][19]).

### Hypercomplex systems.

Let $\Phi$ be a finite hypercomplex system, that is, a finite-dimensional associative algebra with a fixed basis $H = \{ h _ {1} \dots h _ {n} \}$. The algebra $\Phi$ is identified with the space of functions on the finite set $H$, by associating the function $\phi ( h)$ to the element $\sum _ {i=} 1 ^ {n} \phi ( h _ {i} ) h _ {i} \in \Phi$. Let

$$R ^ {x} \phi ( h) = \sum _ { i= } 1 ^ { n } \phi ( h _ {i} ) h _ {i} \star x ,$$

where $h _ {i} \star x$ is the product of $h _ {i}$ and $x$ in the algebra $\Phi$. The operators $R ^ {x}$ form a family of generalized displacement operators if and only if one of the elements of the basis $H$ is a right identity in $\Phi$. In this way a correspondence is established between arbitrary generalized displacement operators in the space of functions on a finite set and finite hypercomplex systems. Thus, the concept of a generalized displacement operator can be regarded as a far-reaching generalization of the classical concept of a hypercomplex system. Important examples of generalized displacement operators that can naturally be treated as hypercomplex systems with a countable basis or with a basis with the power of the continuum are considered in, for example, [4], [5] and [8].

## Generators and Lie's theorems for generalized displacement operators.

Let the hypergroup $H$ be a differentiable (or complex-analytic) manifold and let $u ( x , y ) = R ^ {x} \phi ( y)$ be a differentiable (respectively, holomorphic) function on $H \times H$ for all $\phi \in \Phi$. Let $( h _ {1} \dots h _ {n} )$ be the local coordinates of $h \in H$, where the coordinate system is chosen so that the neutral element has coordinates $( 0 \dots 0)$. The generators (infinitesimal operators) of the right displacement of order $k$ for $R ^ {x}$ are the linear operators of the form

$$R _ {k _ {1} \dots k _ {n} ; h } : \phi ( h) \mapsto \left . \frac{\partial ^ {k} u ( x , h ) }{\partial ^ {k _ {1} } {x _ {1} } \dots \partial ^ {k _ {n} } {x _ {n} } } \right | _ {x= 0 } ,$$

where $u ( x , h ) = R ^ {x} \phi ( h)$, $k = k _ {1} + \dots + k _ {n}$. The generators of the left displacement are defined similarly by the equation

$$L _ {k _ {1} \dots k _ {n} ; h } : \phi ( h) \mapsto \left . \frac{\partial ^ {k} u ( h , x ) }{\partial ^ {k _ {1} } {x _ {1} } \dots \partial ^ {k _ {n} } {x _ {n} } } \right | _ {x= 0 } .$$

From the associativity axiom it can be deduced that any left displacement generator commutes with all right displacement generators (as well as with the operators $R ^ {x}$). Differentiation of the associativity condition $R ^ {x} L ^ {y} \phi ( h) = L ^ {y} R ^ {x} \phi ( h)$ with respect to $h _ {1} \dots h _ {n}$ an appropriate number of times and putting $h= 0$ gives the system of equations

$$\tag{* } L _ {k _ {1} \dots k _ {n} ; x } ( u) = R _ {k _ {1} \dots k _ {n} ; y } ( u) ,$$

where $u ( x , y ) = R ^ {x} \phi ( y)$. This system should be regarded as the generalization of Lie's first direct theorem to the case of generalized displacement operators (see [3] for Delsarte generalized displacement operators and [5] for the general case). Not all the equations (*) need necessarily be involved in determining $u ( x , y )$. For example, for displacements on a Lie group, the generators of the first order already uniquely determine the function $u$( that is, the group multiplication). In the general case certain generators of lower orders may degenerate, for example, into multiplication by a constant, so that the corresponding equations in (*) contain no useful information. This gives rise to the following problem: Select a minimal number of equations from (*) that uniquely determine the generalized displacement operators. Here, degenerate generators increase the number of initial conditions. If a finite system of the form (*) under certain initial conditions, including the condition $u \mid _ {x=} 0 = \phi ( y)$, uniquely determines the solution $u ( x , y ) = R ^ {x} \phi ( y)$, and if the operators on the left-hand side of the system commute with all the operators on the right-hand side, then the operators $R ^ {x}$ are generalized displacement operators. This assertion is the analogue of Lie's first converse theorem [5]. Analogues of Lie's second and third (direct and converse) theorems have been proved for a certain class of generalized displacement operators (see [5]). In particular, generalized displacement operators have been constructed on the space of infinitely-differentiable functions in $n$ variables, for which the right (left) displacement generators generate any given $n$- dimensional Lie algebra. An explicit description of these generators has been obtained in the form of integro-differential operators of the second order [10]. By means of similar techniques one has constructed generators of any order [11] acting on the space of entire analytic functions (in $n$ variables) and generating any $n$- dimensional real Lie algebra; the generalized displacement operators can be re-established in terms of these generators. Generalized displacement operators can be constructed not only starting from Lie algebras, but also from commutation relations of a more general kind (see [7], [12]). Thus, generalized displacement operators on the line were already constructed in [1], starting from an explicitly-given second-order operator by means of a series, analogous to Taylor series, which gives an expansion of the ordinary displacement in powers of the differentiation operator. Commutative generalized displacement operators on the line with a second-order generator of Sturm–Liouville type have been described in detail (see [4] and [5]) and applications have been indicated to Sturm–Liouville operators and equations. A complete classification has been given [14] of generalized displacement operators with a generator of Sturm–Liouville type (including non-commutative ones) on the space of differentiable functions on the line.

## Representations of generalized displacement operators and hypergroup algebras.

Representation theory is not so well developed for generalized displacement operators as it is for groups, but it is constructed in a similar way. This analogy extends quite far; for example, with the aid of the concept of a generator, representations of generalized displacement operators can be studied by an infinitesimal method, just as this is done for Lie groups (see [3], [5], [11]). It is convenient to treat representations of generalized displacement operators as representations of associative hypergroup algebras, analogous to group algebras (see Infinite-dimensional representation). If the hypergroup $H$ is locally compact, then the space $M ( H)$ of complex Radon measures on $H$ with compact support acquires the structure of a hypergroup algebra with respect to the generalized convolution $f \star g$ of elements $f , g \in M ( H)$, defined by the equation

$$\int\limits \phi d ( f \star g ) = \int\limits \int\limits R ^ {y} \phi ( x) d f ( x) d g ( y) ,$$

where $\phi$ is an arbitrary continuous function on $H$. The structure of a hypergroup can be defined similarly on the space $D ( H)$ of generalized functions of compact support on $H$( or on the space $A ( H)$ of analytic functionals on $H$), if the hypergroup $H$ is a differentiable (complex-analytic) manifold. In the natural topologies, $M ( H)$, $D ( H)$ and $A ( H)$ are topological algebras, and the $\delta$- function concentrated at the neutral element $e \in H$ is the right identity in each of these algebras. The converse is also valid: If in the space $M ( H)$( respectively, $D ( H)$, $A ( H)$), endowed with the natural topology, the structure of a topological associative algebra with a right identity that is a $\delta$- function has been given, then there exists a (unique) hypergroup structure on $H$ such that the generalized convolution is the same as multiplication in $M ( H)$( respectively, in $D ( H)$ or $A ( H)$). Continuous representations of the algebra $M ( H)$( $D ( H)$, $A ( H)$) can be interpreted as continuous (infinitely-differentiable, holomorphic) representations of the corresponding generalized displacement operators [20].

The theory of symmetric representations of Banach hypergroup algebras with an involution is an analogue of the theory of unitary representations of groups. The most complete results (see [4]–) have been obtained for representations of commutative and compact generalized displacement operators. Under certain conditions the space $L _ {1} ( H , m )$ of functions on $H$ that are summable with respect to a positive measure $m$ can be endowed with the structure of a Banach hypergroup algebra with an involution. One of these conditions is that the measure $m$ be invariant under generalized displacements (for different versions of the precise definitions see [4]–, [15][19]). Under natural assumptions, the uniqueness (up to a scalar multiple) has been proved of a measure that is invariant under right or left generalized displacements; there are also sufficient conditions for the existence of such a measure (conditions like compactness, commutativity or discreteness of the hypergroup, see [8], [16][18]). However, the problem of the existence of an invariant measure for generalized displacement operators of general form remains open (1982). Along with $L _ {1} ( H , m )$, an important role is played by a Banach hypergroup algebra of measures of bounded variation and a hypergroup $C ^ {*}$- algebra.

Banach hypergroup algebras and their symmetric representations have been studied in [4],

[8], [15][19]. Algebras of analytic functionals related to certain generalized displacement operators on the line were studied in . For generalized displacement operators of general type, topological hypergroup algebras and their representations were considered in [20], where problems of spectral analysis and synthesis were treated as problems in the study of ideals of hypergroup algebras. The technique of hypergroup algebras was applied in [12] to solve problems in mathematical physics in the framework of the operator methods of V.P. Maslov.

## Harmonic analysis.

The following model reveals the structure of commutative generalized displacement operators (see [4], [5]). Let two positive measures $m _ {1}$ and $m _ {2}$ be given on spaces $H _ {1}$ and $H _ {2}$, respectively, and, with the aid of a function $\chi ( x , y )$ defined on $H _ {1} \times H _ {2}$, let a generalized Fourier transformation

$$\phi ( x) \mapsto f ( y) = \int\limits \phi ( x) \overline{ {\chi ( x , y ) }}\; \ d m _ {1} ( x)$$

be given, which defines an isomorphism of the Hilbert spaces $L _ {2} ( H _ {1} , m _ {1} )$ and $L _ {2} ( H _ {2} , m _ {2} )$. Assume that the inversion formula

$$\phi ( x) = \int\limits f ( y) \chi ( x , y ) d m _ {2} ( y)$$

is valid. If the measure $m _ {2}$ is discrete, then this formula gives an expansion of $\phi ( x)$ in a generalized Fourier series. It turns out that $H _ {1}$ has the structure of a hypergroup if for some $e \in H _ {1}$ and all $y \in H _ {2}$ one has $\chi ( e , y ) = 1$. In this case generalized displacement operators are defined by the formula

$$R ^ {k} \phi ( x) = \int\limits f ( y) \chi ( k , y ) \chi ( x , y ) d m _ {2} ( y) .$$

Hence, generalized displacement operators arise naturally in problems concerned with expansion in orthogonal systems of functions, in the spectral theory of operators, etc., which ensure a wide range of applications of the theory of generalized displacement operators (see, for example, [4], [5], [8], [15][19]). In the case of commutative generalized displacement operators, Bochner's theorem on representations of positive-definite functions and the law of Pontryagin duality have been generalized, a generalized Fourier transformation has been defined, and an analogue of the Plancherel theorem has been proved

(for various versions of these results see [15][17]). With the aid of representation theory, harmonic analysis may also be constructed for non-commutative generalized displacement operators. E.g., non-commutative analogues of the Plancherel theorem and the inversion formula have been obtained for generalized displacement operators, which include as a special case the corresponding results for locally compact groups; an analogue of the Peter–Weyl theorem is valid for compact generalized displacement operators. Versions of harmonic analysis in the spirit of the theory of almost-periodic and mean-periodic functions have been considered for generalized displacement operators (see [1], [2], [4], [8], , [14], [20]). An analogue of Wiener's Tauberian theorem has been obtained for commutative generalized displacement operators (cf. Wiener Tauberian theorem), and questions of spectral synthesis have been considered (see [21], [22]). For applications of generalized displacement operators to harmonic analysis on groups see [8], [13], [16], [19].

#### References

See also [a2]. If the pair of a locally compact group $G$ and a compact subgroup $K$ forms a Gel'fand pair, then the generalized displacement operators associated with it commute. With expansions and dual expansions in Jacobi polynomials a commutative hypergroup structure can be associated. See [a1] for a class of Sturm–Liouville operators yielding generalized displacement operators.