Some Properties About Norm Of Generalized Lyapunov Operator

As a strong and powerful tool in the research of theory and applications of modern mathematics, the matrix, which has been discussed from the view of algebra and analysis, is playing more and more important role in the fields of mathematics itself, physics, economics, etc.The linear matrix operator, which is a crucial composing of matrix theory and concentrated on linear matrix equations in the past, is not taken too seriously on itself. The core problem of research on the linear matrix operator is that whether the maximal operator norm and minimal operator norm of the linear matrix operator equal to those of the operator restricted to symmetric subspace.This paper discusses the problem of maximal and minimal operator norm of generalized Lyapunov operator(?)(X)=AXB+BTXAT,(?)X∈Mn×n (R) in Mn with the Frobenius matrix norm, and proves firstly that when A∈Mn is a tridiagonal matrix and B∈Mn is a diagonal matrix, the maximal operator norm problem is true, meanwhile provides some counterexamples of the generalization of the problem and a conclusion that these counterexamples emerge sparsely, and then affirms that the problem is right for n=2; and secondly, gives counterexample of the minimal operator norm problem with general case of n≥3and the result that the frequency of counterexample emerging is small; Finally, This paper gets a proof of the maximal operator norm for (?)(X)=AX+XAT,(?)X∈Mn×n (R) with the cases of n=3and tridiagonal matrix A, however, which is an unsolved problem with generalized condition of n≥4.