The paper studies Sylvester and Lyapunov operators in real and complex matrix spaces including operators arising in the theory of linear time-invariant systems. The linear operator \({\mathcal M}:{\mathbb F}^{m\times n}\rightarrow {\mathbb F}^{p\times q}\) (with \({\mathbb F}={\mathbb R}\) or \({\mathbb C}\)) is elementary if \({\mathcal M}[X]=AXB\) for some \(A\in {\mathbb F}^{p\times m}\) and \(B\in {\mathbb F}^{n\times q}\). Each \({\mathcal M}\) can be represented as a sum of a minimal number (called the Sylvester index \(Sy[{\mathcal M}]\) of \({\mathcal M}\)) of elementary operators. An expression for \(Sy[{\mathcal M}]\) is given using a special permutation operator \({\mathcal V}_{p,m}: {\mathbb F}^{pq\times mn}\rightarrow {\mathbb F}^{pm\times nq}\) such that the image \({\mathcal V}_{p,m}(B^T\otimes A)\) of the matrix of a non-zero elementary operator is equal to the rank 1 matrix vec\([A]\)row\([B]\) (vec\([X]\) and row\([X]\) are the column-wise and row-wise vector representations of the matrix \(X\)). The application \({\mathcal V}_{p,m}\) reduces a sum of Kronecker products of matrices to the standard product of two matrices.
A linear operator \({\mathcal L}:{\mathbb F}^{n\times n}\rightarrow {\mathbb F}^{n\times n}\) is a Lyapunov operator (LO) if \(({\mathcal L}[X])^*={\mathcal L}[X^*]\) (A) the star denoting transposition in the real case and complex conjugation and transposition in the complex case. Characterizations and parametrizations of the sets of real and complex LOs are given and their dimensions are found; Lyapunov indices for LOs are introduced and calculated. Similar results are given for some classes of Lyapunov-like linear and pseudo-linear operators in which condition (A) is replaced by a similar one. The notion of Lyapunov singular values of an LO is introduced and the application of these values to the sensitivity and a posteriori error analysis of Lyapunov equations is discussed.