AbstractLet X, Y be linear spaces over a field $${\mathbb {K}}$$
K
. Assume that $$f :X^2\rightarrow Y$$
f
:
X
2
→
Y
satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all $$x,x_i,y,y_i \in X$$
x
,
x
i
,
y
,
y
i
∈
X
and with $$a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}$$
a
i
,
b
i
∈
K
\
{
0
}
, $$A_i,\,B_i \in {\mathbb {K}}$$
A
i
,
B
i
∈
K
($$i \in \{1,2\}$$
i
∈
{
1
,
2
}
). It is easy to see that such a function satisfies the functional equation for all $$x_i,y_i \in X$$
x
i
,
y
i
∈
X
($$i \in \{1,2\}$$
i
∈
{
1
,
2
}
), where $$C_1:=A_1B_1$$
C
1
:
=
A
1
B
1
, $$C_2:=A_1B_2$$
C
2
:
=
A
1
B
2
, $$C_3:=A_2B_1$$
C
3
:
=
A
2
B
1
, $$C_4:=A_2B_2$$
C
4
:
=
A
2
B
2
. We describe the form of solutions and study relations between $$(*)$$
(
∗
)
and $$(**)$$
(
∗
∗
)
.
Spectral properties of some unions of linear spaces
