1.Introduction

Let *M* be a monoid i.e., is a semigroup with an identify element that we denote by e and *σ,τ: M → M* are two involutive automorphisms. That is *σ*(*xy*) = *σ*(*x*)*σ*(*y*), *τ*(*xy*) *= τ*(*x*)*τ*(*y*) and *σ*(*σ*(*x*)) = *x*, *τ*(*τ*(*x*)) = *x* for all *x,y* ∈ *M*. By a variant of Wilson's functional equation on *M* we mean the functional equation

(1.1) *f*(*xσ*(*y*)) + *f*(*τ*(*y*)*x*) = 2*f*(*x*)*g*(*y*), *x*,*y* ∈ *M*,

Where *f, g:M*→ **C** are the unknown functions. A special case of Wilson's functional equation is d'Alembert's functional equation:

(1.2) *f*(*xσ*(*y*)) + *f*(*τ*(*y*)*x*) = 2*f*(*x*)*f*(*y*), *x,y* ∈ *M*

The solutions of equation (1.2) are known(^{2}). Further contextual and historical discussion on the functional equation (1.1) and (1.2) can be found, e.g., in (6.2).

The present paper studies an extension to a situation where the unknown functions f, g map a possibly non-abelian group or monoid into a complex Hilbert space *H* with the Hadamard product. Our considerations refer mainly to results by Rezaei (^{4}), Zeglami (^{11}).

It has been proved (^{3}) that the functional equation (1.2) with *σ = id* is superstable in

the class of functions *f:G* → **C**, if every such function satisfies the inequality

|*f*(*xy*) + *f*(*τ*(*y*)*x*) - 2*f*(*x*)*f*(*y*)| ≤ *ϵ*
*f* or all *x,y* ∈ *G*,

where *ϵ* is a fixed positive real number. Then either *f* is a bounded function or

*f*(*xy*) + *f*(*τ*(*y*)*x*) = 2*f*(*x*)*f*(*y*), *x,y* ∈ *G.*

Let *H* be a separable Hilbert space with a orthonormal basis {_{
en
} , *n* ∈ **N**}. For two vectors *x,y* ∈ *H*, the Hadamard product, also known as the entrywise product on the Hilbert space *H* is defined by

The Cauchy-Schwarz inequality together with the Parseval identity ensure that the Hadamard multiplication is well defined. In fact,

The purpose of this work is first to give a characterization, in terms of multiplicative functions, the solutions of the Hilbert space valued functional equation by Hadamard product:

(1.5) *f*(*xσ*(*y*)) + *f*(*τ*(*y*)*x*) = 2*g*(*x*) * *f*(*y*), *x,y* ∈ *M.*

When f we determine the solutions of the functional equation

(1.6) *f*(*xσ*(*y*)) + *f*(*τ*(*y*)*x*) = 2*f*(*x*) * *g*(*y*), *x,y* ∈ *M,*

where *f,g:M* → *H* are the unknown functions. Second, we determine a characterization of the following d'Alembert-Hilbert-valued functional equation:

(1.7) *f*(*xσ*(*y*)) + *f*(*τ*(*y*)*x*) = 2*f*(*x*) * *f*(*y*), *x,y* ∈ *M.*

Throughout the paper, **N, R** and **C** stand for the sets of positive integers, real numbers and complex numbers, respectively. We let *G* denote a group and *S* denote a semigroup i.e., a set with an associative composition rule.

A function *A:M* → **C** is called additive, if it satisfies *A*(*xy*) = *A*(*x*) + *A*(*y*) for all *x,y* ∈ *M.*

A multiplicative function on *M* is a map *χ:M* → **C** such that *χ(xy) = χ(x) χ(y)* for all *x,y* ∈ *M.*

A monoid *M* is generated by its squares if for every
for some _{
x1,x2,…,xn
} ∈ *M*.

A character on a group *G* is a homomorphism from *G* into the multiplicative of non-zero complex numbers. While a non-zero multiplicative function on a group can never take the value 0, it is posible for a multiplicative function on a monoid *M* to take the value 0 on a proper, non-empty subset of *M*. If *χ:M →*
**C** is multiplicative and *χ ≠* 0, then

_{
Iχ =
} {*x* ∈ *M* / *χ(x) =* 0}

is either empty or a proper subset of *M*. The fact that *χ* is multiplicative establishes that _{
Iχ
}

is a two-sided ideal in *M* if not empty (for us an ideal is never the empty set). It follows also that *M*\_{
Iχ
} is a subsemigroup of *M*.

Let *C*(*M*) denote the algebra of continuous functions from *M* into **C**.

2.Solutions of (1.5) and (1.6)

In this section, we solve the functional equation (1.5) by expressing its solutions in terms of

multiplicative functions.

**Theorem 2.1.** Let *M* be a monoid, let *σ, τ:M* → *M* be involutive automorphisms. Assume

that the functions *f, g:M* → *H* satisfy (1.5). Then, there exists a positive integer *N* such that

for all *x* ∈ *M* and *k* > 0. Furthermore, for every *k* ∈ {1,2,….,*N*}, we have the following possibilities:

for all *x* ∈ *M,* where _{
χk
} is a non-zero multiplicative function of *M* such that _{
χk °
}
*σ* ° *τ* =_{
χk
} ° *τ* ° *σ* and α_{
k
} ∈ **C** \ {0}. If *M* is a topological monoid and *f* ∈ *C*(*M*), then _{
χk, χk ° σ ° τ
} ∈ *C* (*M*)

**Proof.** For every integer *k* ≥ 0, consider the functions *f*
_{k},*g*
_{k}:*M* → **C** defined by

If we put *y = e* in (2.1), we find that _{
fk
} (*x*) = _{
fk
} (*e*)_{
gk
} (*x*). So, if we take α_{k} = *f*
_{k}(*e*),

equation (2.1) can be written as follows:

_{
αkgk
} (*xσ*(*y*)) + _{
αkgk
} (*τ*(*y*)*x*) = 2_{
αkgk
} (*y*) *f* or all *x,y* ∈ *M.*

Then, either α_{k} = 0 or *g*
_{k} is a solution of equation (1.6). In view of (^{(2)}), Theorem 3.2), one of the following statements holds:

(a) We have that

*f*
_{k} = 0 and *g*
_{k} is an arbitrary function.

(b) There exists a multiplicative function *χ*
_{k} such that

If *H* is infinite-dimensional, then

for every *x* ∈ *M.* Since *g*
_{k}(*e*) = 1, statement (b) is not possible for infinitely many positive integers *k*. Hence, there exists some positive integer *N* such that *f*
_{k} = 0 for every *k* > *N*. Thus, *g*
_{k} is an arbitrary function for any *k* > *N,* f can be represented as

and the expressions of the component functions *f*
_{n} and *g*
_{n}, 1 ≤ *n* ≤ *N*, of f and g come from statements (a) and (b) above. In the case where *H* is finite- dimensional, the proof is clear.

As a consequence of Theorem 2.1 we derive formulas for the solutions of d'Alembert's Hilbert space valued functional equation (1.7).

**Corollary 2.2.** Let *M* be a monoid, let *σ,τ*:*M* → *M* be involutive automorphisms. Assume

that the functions *g*:*M*→ *H* satisfy (1.7). Then, there exists a positive integer *N* such that

for all *x* ∈ *M* and *k* > 0. Furthermore, for every *k* ∈ {1,2,….,N}, such that

where ϵ_{k} = 1 or 0 for every *k* ∈ {1,2,…..,*N*}. for all *x ∈ M*, where *χ*
_{k} is a non-zero multiplicative function of *M* such that *χ*
_{k} ° *σ* ° *τ* = _{
χk
} ° *τ* ° *σ*

If *M* is a topological monoid and *f* ∈ *C*(*M*), then _{
χk, χk °
}
*σ* ° *τ* ∈ *C*(*M*).

**Proof.** The proof follows by putting *f = g* in Theorem 2.1.

**Corollary 2.3.** Let *M* be a monoid, let *τ*: *M* → *M* be involutive automorphisms. Assume that

the functions *f*, *g*; *M* → *H* satisfy

*f*(*xt*) + *f*(*τ*(*y*)*x*) = 2*g*(*x*) * *f*(*y*).

Then, there exists a positive integer *N* such that

for all x ∈ *M* and *k* > 0. Furthermore, for every *k* ∈ {1,2,….,N}, we have the following possibilities:

for all x ∈ *M,* where _{
χk
} is a non-zero multiplicative function of *M* and _{
αk
} ∈ **C** \ {0}.

If *M* is a topological monoid and *f* ∈ *C*(*M*), then _{
χk, χk ° τ
} ∈ *C*(*M*).

**Proof.** The proof follows by putting *σ = id* in Theorem 2.1.

We complete this section with a result concerning Wilson Hilbert space valued functional equation (1.6).

**Theorem 2.4.** Let *M* be a monoid which is generated by its squares, let *σ,τ*: *M* → *M* be involutive automorphisms. Assume that the pair *f,g: M* → **C**, satisfy Wilson's Hilbert valued functional equation (1.6). Then, there exists a positive integer *N* such that

for all x ∈ *M* and *k* > 0. Furthermore, for every *k* ∈ {1,2,….,*N*}, we have the following possibilities:

Where _{
χk: M
} → **C** is a non-zero multiplicative function with _{
χk °
}
*σ* ° *τ* = _{
χk °
}
*τ* ° *σ* and for some α_{k} ∈ **C** \ {0}.

(ii) There exists a non-zero multiplicative function _{
χk: M
} → **C** with with _{
χk °
}
*σ* ° *τ* = _{
χk °
}
*τ* ° *σ* such that

Furthermore, we have

(1) If _{
χk ≠ χk °
}
*σ* ° *τ,* then

for some _{
αk
} , β_{k} ∈ **C** \ {0}.

(2) If _{
χk = χk °
}
*σ* ° *τ,* then there exists a non-zero additive function A_{k}: *M* \ _{
Iχk°σ →
}
**C** with *A*
_{k} ° τ = -*A*
_{k} ° *σ* such that

for some _{
αk
} , ∈ **C.**

Conversely, if *f* and *g* have the forms described above, then the pair (*f,g*) is a solution of equation (1.6). Moreover, if *M* is a topological monoid generated by its squares, and *f, g* ∈ *C* (*M*), then _{
χk, χk ° σ, χk ° τ, χk ° σ ° τ
} ∈ *C*(*M*), while *A*
_{k} ∈ *C*(*M*\I_{
χk°σ
} ).

**Proof.** We proceed as in the proof of Theorem 2.1. For every integer *k* ≥ 0, we consider the functions *f*
_{k},*g*
_{k}:*M* → **C**, defined by

Since the pair (*f,g*) satisfies (1.6), for all *k* ∈ **N** we have

By (^{6}),Theorem 3.4) we infer that there are only the following cases

for some _{
αk
} , ∈ **C.** Conversely, the functions given with properties satisfy the functional equation (2.2). The continuation of the proof depends on the dimension of *H*. In fact, if *H* is infinite-dimensional, then

⟨(g)/x),ex⟩ = gk(x) → 0 as k → + ∞

for every *x* ∈ *M*. Statements (b) and (c) are not possible for infinitely positive integers n. Hence, there exists some positive integer *N* such that *f*
_{k} = 0 for every *k* > *N*. Thus, f can be represented as

*g*
_{k} is an arbitrary function for any *k* > *N*, and expressions of the component functions *f*
_{n} and _{
gn
} , 1 ≤ *n* ≤ *N* of *f* and *g* follow from the previous discussion. In the case where *H* is a finite-dimensional space, the proof is clear.

**Corollary 2.5.** Let *M* be a monoid which is generated by its squares, let *τ*:*M* → be an involutive automorphism, and let the pair *f*,*g*: *M* → *H* satisfy the functional equation

*f*(*xy*) + *f*(*τ*(*y*)*x*) = 2*f*(*x*) * *g*(*y*), *x, y* ∈ *M*.

Then, there exists a positive integer *N* such that

for all *x* ∈ *M* and *k* > 0. Furthermore, for every *k* ∈ {1,2,….,*N*} we have the following possibilites:

(2) If _{
χk = χk
} ° τ, then there exists an additive function A_{k}:*M*\I_{
χk
} → **C** with A_{k} ° τ = -A_{k} such that

for some α_{k} ∈ **C**.

Conversely, if *f* and *g* have the forms described above, then the pair (*f,g*) is a solution. Moreover, if *M* is a topological monoid generated by its squares, and *f,g* ∈ *C*(*M*), then _{
χk, χk
} ° τ ∈ *C*(*M*), while *A*
_{k} ∈ *C*(_{
M\Iχk
} ).

**Proof**. The proof follows by putting *σ* = *id* in Theorem 2.4.

3.Superstability of Hilbert valued cosine type functional equations

The main result of this section is Theorem 3.3 that contains a superstability result for the functional equation (1.6). For the proof of our result we will begin by pointing out a superstability result for the equation

(3.1) *f*(*xy*) + *f*(*σ*(*y*)*x*) = 2*f*(*x*)*g*(*y*)

where *f*, *g*:*G* → **C** are the unknown functions.

**Proposition 3.1.** Let δ > 0 be given, let *M* be a monoid and let *σ* is an involutive morphism of *M*. Assume that the functions *f,g*:*M* → **C** satisfies the inequality

and that *g* is unbounded. Then, the ordered pair (*f, g*) satisfies equation (3.1).

**Proof.** The proof is part of the proof of (^{3} ,Theorem 2.1 and Theorem 3.7) if we put χ = 1 that deals with *M* being a group.

**Corollary 3.2.** Let δ > 0 be given and let *G* be a monoid. Assume that the function *f*:*G* → **C** satisfies the inequality

where *μ* is a multiplicative function.

**Proof.** The proof follows immediately from Propositon 3.1 and Theorem (^{1}, Theorem 4).

**Theorem 3.3.** Let δ > 0 be given and let *M* be a monoid. Assume that the functions *f, g*: *M* → *H* satisfy the inequality

(3.2) ||*f*(*xy*) + *f*(*σ*(*y*)*x*) - 2*f*(*x*)**g*(*y*)||≤ δ *f* or all *x,y* ∈ *M*.

Then, either

(i) there exists *k* ≥ 1 such that the function *x* ↦ ⟨*g*(*x*), _{
ek
} ⟩ is bounded, or

(ii) the pair (*f, g*) is a solution of the functional equation:

(3.3.) *f*(*xy*) + *f*(*σ*(*y*)*x*) = 2*f*(*x*) * *g*(*y*).

**Proof.** Suppose that the pair (*f, g*) satisfies (3.2). By applying the Parseval identity and the definition of Hadamard product with the inequality (3.2), we find that the scalar valued functions *f*
_{k}, *g*
_{k} defined by

*f*
_{k}(*x*) = ⟨*f*(*x*), _{
ek
} ⟩ and _{
gk
} (*x*) = ⟨*g*(*x*), _{
ek
} ⟩ f or *x* ∈ *M,*

satisfy the inequality

|*f*
_{k}(*xy*) + *f*
_{k}(*σ*(*y*)*x*) - 2*f*
_{k}(*x*)*g*
_{k}(*y*)| ≤ δ *f* or all *x,y* ∈ *M.*

According to Proposition 3.1, for all *k* ∈ **N**, we have that either the function *x* ↦ ⟨*g*(*x*), _{
ek
} ⟩ is bounded or the pair (_{
fk
} , _{
gk
} ) is a solution of (3.1). Then, we conclude that the pair (*f, g*) satisfies equation (3.3) if assertion (i) fails.

In (^{(4)}) it was proved that if *g:*
*H* → *H* is surjective, then every component function *x* ↦ ⟨*g*(*x*), _{
en
} ⟩

is unbounded. By applying Theorem (3.3), this leads to the following result.

**Corollary 3.4.** Let δ > 0 be given. Assume that functions *f*, *g:*
*H* → *H,* where g is surjective, satisfy the inequality

||*f*(*xy*) + *f*(*σ*(*y*)*x*) - 2*f*(*x*)**g(y)*|| ≤ δ for all *x,y* ∈ *H.*

Then, the pair (*f, g*) satisfies the equation

*f*(*xy*) + *f*(*σ*(*y*)*x*) = 2*f*(*x*)**g*(*y*) for all *x,y* ∈ *H.*

**Proof.** Since g is surjective, then every component function *x* ↦ ⟨*g*(*x*), _{
en
} ⟩ is unbounded. Thus, the proof follows immediately from Theorem 3.3.

**Corollary 3.5.** Let δ > 0 be given and let *G* be a topological group. Assume that the function *g:G* → *H* satisfies the inequality

||*g*(*xy*) + *g*(*σ*(*y*)*x*) - 2*g*(*x*) * *g*(*y*)|| ≤ δ *f* or all x,y ∈ G.

Then, either there exists *k* ≥ 1 such that

or there exist a multiplicative function *χ*
_{k}: *M* → **C** \ {0} and a positive integer *N* such that

where ϵ_{n} = 1 or 0 for every *n* ∈ {1,2,….., *N*}.

**Proof.** If we put *f* = *g* in Theorem 3.3, we immediately have that either there exists *k* ≥ 1 such that the function x ↦ ⟨*g*(*x*),*e*
_{k}⟩ is bounded or *g* is a solution of the equation

*g*(*xy*) + *g*(*σ*(*y*)*x*) = 2*g*(*x*) * *g*(*y*), *x,y* ∈ *G*.

The remainder of the proof follows if we put χ = 1 from Corollary (^{3}), Corollary 3.8) and Corollary 2.3.

**Corollary. 3.6.** Let δ > 0 be given and let *G* be a group with identity element. Let *g:G* → *H* such that

||*g*(*xy*) + 2*g*(*x*) * *g*(*y*)|| ≤ δ *f* or all x,y ∈ *G*. Then either *g* is bounded or *g* is multiplicative.

**Proof.** From Corollary 2.2 and Corollary 2.5 and then using (^{3}, Corollary 3.9).