integroidummat

Integroidummat is a theoretical construct used in discussions of linear algebra and operator theory to denote a matrix-like array whose entries are integro-differential operators or integral kernels acting on function spaces. The term is not part of a formal standard, but it is employed in pedagogical or exploratory contexts to illustrate how a finite matrix structure can be built from operator-valued entries.

Definition and structure: A square integroidummat of size n over a field F consists of entries Aij, with i and j ranging from 1 to n, where each Aij is an operator that maps a suitable function space V (for example, a subspace of L2 or C0) to another function space. Matrix addition is performed entrywise, and multiplication is defined by the usual matrix rule with operator composition: (AB)ij = sum_k Aik Bkj, where the products Aik Bkj are interpreted as the composition of operators and any integral transforms involved are applied in the natural order. When viewed as a single operator on Vn, an integroidummat acts by applying the row-wise combination of operators to an n-tuple of functions.

Relation to known concepts: Integroidummats generalize ordinary matrices (when all entries reduce to scalar multiplications) and matrices of operators (when each entry is a linear operator). They also connect to operator-valued matrices used in the study of coupled systems of integral and differential equations. The determinant concept generalizes to operator determinants in appropriate settings, such as Fredholm determinants for certain classes of integral operators.

Examples and applications: A simple 2×2 example might include A11 as an integral operator (f ↦ ∫0^1 f(s) ds), A12 as a scalar multiple of the identity, A21 as a differentiation operator (f ↦ f′), and A22 as another linear operator. Integroidummats arise in the modeling and discretization of systems combining integral and differential relationships, in the representation of coupled integral equations, and in the spectral analysis of operator systems.

See also: matrix of operators, operator-valued matrix, integro-differential equation, Fredholm operator.