Norm Estimates for Solutions of Polynomial Operator Equations

where c 0 = 1, c k ∈ C (k = 1, . . . , m), A, B, C are given linear bounded operators in a Banach spaceE with a norm ‖ ⋅ ‖ and X is to be found. There is a long tradition of finding different expressions for the solution of an operator equation in the form of operator integrals and series, some prominent examples of which occur in the works of E. Heinz, M. G. Krein, M. Rosenblum and R. Bhatia, C. Davis, and A. McIntosh. A comprehensive summary of these is contained in [1] and [2]. This tradition was continued in [3], which deals with the equation


Introduction and Statement of the Main Result
The recently suggested gradient based iterative methods are more powerful; compare [10][11][12][13] and references therein.
The aim of the present paper is to derive representations of solutions to (1) and (2) and to estimate the norms of these solutions.Such estimates are important, in particular, for the investigations of linear and perturbations of nonlinear differential and difference equations.Norm estimates for solutions of the Sylvester equation whose coefficients are normal operators can be found in [2].In the finite dimensional case solution estimates for (4) and (6) have been established in [14].
In this paper, in particular, we considerably generalize the main results from [14].
Denote by () and   () the spectrum and spectral radius of , respectively.
Then ( 1) has a unique solution  which can be represented by and the series strongly converges.
The proof of this theorem is presented in the next section.
Lemma 2. Under condition ( 14) equation ( 1) has a unique solution defined by Furthermore, for  ∈ C, satisfying consider the operator The series converges and We thus get the following result.

Proof of Theorem 1.
Put Lemma 2 implies  =   .Due to the previous lemma   are solutions to the equations Continuing this process for  = 3, . . ., , according to Lemma 2, we prove the theorem.

Representations of Solutions to (2)
Assume that  is invertible; then from (2) we have and the series strongly converges.
Condition (27) does not allow us to consider the Lyapunov equation, since   () =   ( * ).Because of this we are going to derive the representation of solutions to (2) under other conditions.To this end put According to (19) for a solution to (26) we have provided Here  −1  is defined as Now the invertibility condition for  can be removed.We thus have proved the following result.Equation ( 4) is equivalent to the following one: Due to Theorem 9.2 from [2] a solution of (36) is defined by the equality as claimed.

Journal of Mathematics
Corollary 7. Let the condition hold.Then the equation has a unique solution   , which can be represented as A solution of (39) is also given by   = (  −   ) −1 .So under condition (38) we have Now assume that Then by ( 41) Then  =   .Due to the previous lemma   are solutions to the equations provided (42) holds.So Continuing this process for  = 3, . . ., , according to Lemma 5, we obtain We arrive at the following result.

Solution Estimates in the Finite Dimensional Case
4.1.Equation (1).In this section E = C  is a Euclidean space.
If both  and  are normal, then For the Sylvester equation we have the following result.
Corollary 15.Let conditions ( 9) and ( 77 By virtue of Corollary 15 we can assert that a unique solution   of (7) satisfies the inequality provided  satisfies conditions (77) and   () < 1.From Corollary 16 it follows that According to Corollary 17 we get the following result.
Then the solution   to the Lyapunov equation (5)