Estimates for functions of two commuting infinite matrices and applications (original) (raw)
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Norm estimates for functions of two commuting matrices
Electronic Journal of Linear Algebra, 2005
Matrix valued analytic functions of two commuting matrices are considered. A precise norm estimate is established. As a particular case, the matrix valued functions of two matrices on tensor products of Euclidean spaces are explored.
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A class of matrix valued analytic functions of two non-commuting matrices is considered. A sharp norm estimate is established. Applications to matrix and differential equations are also discussed.
Norm Estimates for Functions of Two Commuting
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Matrix valued analytic functions of two commuting matrices are considered. A precise norm estimate is established. As a particular case, the matrix valued functions of two matrices on tensor products of Euclidean spaces are explored.
Estimates for Solutions of Differential Equations in a Banach Space via Commutators
Nonautonomous Dynamical Systems, 2018
In a Banach space we consider the equation dx(t)/dt = (A + B(t))×(t) (t ≥ 0), where A is a constant bounded operator, and B(t) is a bounded variable operator.Norm estimates for the solutions of the considered equation are derived in terms of the commutator AB(t) − B(t)A. These estimates give us sharp stability conditions. Our results are new even in the finite dimensional case.We also discuss applications of the obtained results to a class of integro-differential equations.
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Acta Applicandae Mathematicae, 1993
AbstracL A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators. From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved. From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.
OPERATOR LIPSCHITZ ESTIMATE FUNCTIONS ON BANACH SPACES
Journal of Semigroup Theory and Applications, 2019
In this paper, let X, Y be Banach spaces and let ℒ(X, Y) be the space of bounded linear sequence of operators from X toY. We develop the theory of double sequence of operators integrals on ℒ(X, Y) and apply this theory to obtain commutator series estimates, for a large class of functions í µí± í µí± , where í µí°´íµí°´í µí± ∈ ℒ(í µí±), B í µí± ∈ ℒ(í µí±) are scalar type the sequence of operators and í µí± ∈ ℒ(í µí±, í µí±). In particular, we establish this estimate for í µí± í µí± (1 + í µí¼): = |1 + í µí¼| and for diagonalizable estimates derive hold for diagonalizable matrices with a constant independent of the size of the sequence of operators on í µí± = ℓ (1+í µí¼) and í µí± = ℓ (1+í µí¼) , for í µí¼ = 0, and X = Y = c 0. Also, we obtain results for í µí¼ ≥ 0, studied the estimate above [1] in the setting of Banach ideals in ℒ(í µí±, í µí±).
Matrix multiplication operators on Banach function spaces
Proceedings of the Indian Academy of Sciences - Section A, 2006
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Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator
Results in Mathematics, 2019
Let A be the generator of an analytic semigroup (e At) t≥0 on a Banach space X , B be a bounded operator in X and K = AB − BA be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator S −1 l , such that ∞ 0 Se At e Bt dt < ∞, and the operator KS −1 l is bounded and has a sufficiently small norm, we show that ∞ 0 e (A+B)t dt < ∞, where (e (A+B)t) t≥0 is the semigroup generated by A + B. In addition, estimates for the supremum-and L 1norms of the difference e (A+B)t − e At e Bt are derived.