Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
In this note we give an elementary proof that an arbitrary convex function can be uniformly appro... more In this note we give an elementary proof that an arbitrary convex function can be uniformly approximated by a convex C ∞-function on any closed bounded subinterval of the domain. An interesting byproduct of our proof is a global equation for a polygonal (piecewise affine) function.
Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asym... more Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.
The paper studies isolated spectral points of elements of Banach algebras and of bounded linear o... more The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that 0 is isolated in the spectrum of a bounded linear operator T if the (not necessarily closed) space M = {x : limn T n x 1/n = 0} is nonzero and complemented by a closed subspace N satisfying T N ⊂ N ⊂ T X.
We study decompositions of a Banach space operator T in the form T = K + Q, where K is compact (o... more We study decompositions of a Banach space operator T in the form T = K + Q, where K is compact (or meromorphic) and where the spectrum of Q is contained in the set of all accumulation points of the spectrum of T. Many known decomposition results of this type are subsumed in our construction. We prove that every Hilbert space operator has the meromorphic decomposition, and obtain an improvement of a result of Laurie and Radjavi on the West decomposition of Riesz operators in a Banach space.
In this paper we obtain a new inequality of Hilbert type for a finite number of nonnegative seque... more In this paper we obtain a new inequality of Hilbert type for a finite number of nonnegative sequences of real numbers from which we can recover as a special case an inequality due to Pachpatte. We also obtain an integral variant of the inequality.
In this paper we continue our investigation of multivariable integral inequalities of the type co... more In this paper we continue our investigation of multivariable integral inequalities of the type considered by Hilbert and recently by Pachpatte by focusing on fractional derivatives. Our results apply to integrable not necessarily continuous functions, and we are able to relax the original conditions to admit negative exponents in the weight functions.
A parameter dependent nonlinear differential-delay equation in a Banach space is investigated. It... more A parameter dependent nonlinear differential-delay equation in a Banach space is investigated. It is shown that if at the critical value of the parameter the problem satisfies a condition of linearized stability then the problem exhibits a stability which is uniform with respect to the whole range of the parameter values. The general theorem is applied to a diffusion system with applications in biology.
In this note we present an integral representation for the Drazin inverse A D of a complex square... more In this note we present an integral representation for the Drazin inverse A D of a complex square matrix A. This representation does not require any restriction on its eigenvalues.
The paper gives generalisations of Hardy's integral inequality that further extend results of Moh... more The paper gives generalisations of Hardy's integral inequality that further extend results of Mohapatra and Russell (Aequationes Math. 28 (1985), 199-207) and other authors by using α-submultiplicative or α-supermultiplicative functions.
For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ ... more For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ C such that T − λI is upper semi-B-Fredholm and ind (T − λI) ≤ 0, and let E a (T) be the set of all isolated eigenvalues of T in the approximate point spectrum σ a (T) of T. We say that T satisfies generalized a-Weyl's theorem if σ SBF − + (T) = σ a (T)\E a (T). Among other things, we show in this paper that if T satisfies generalized a-Weyl's theorem, then it also satisfies generalized Weyl's theorem σ BW (T) = σ(T) \ E(T), where σ BW (T) is the B-Weyl spectrum of T and E(T) is the set of all eigenvalues of T which are isolated in the spectrum of T. 2000 AMS subject classification: 47A53, 47A55
The paper introduces a special type of a Drazin-like inverse for closed linear operators that ari... more The paper introduces a special type of a Drazin-like inverse for closed linear operators that arises naturally in ergodic theory of operator semi-groups and cosine operator functions. The Drazin inverse for closed linear op-erators defined by Nashed and Zhao [30] and in a more general form by Koliha and Tran [21] is not sufficiently general to be applicable to operator semi-groups. The a-Drazin inverse is in general a closed, not necessarily bounded, operator. The paper gives applications of the inverse to partial differential equations.
Rendiconti del Circolo Matematico di Palermo, 1998
We study spectral sets of elements of a Banach algebra in terms of the existence of idempotent pa... more We study spectral sets of elements of a Banach algebra in terms of the existence of idempotent partitions of the unit of the algebra. Circularly isolated spectral sets are characterized, and a new Mbekhta decomposition for bounded linear operators is obtained.
The present paper continues the investigation of the relation between abstract ergodic theorems a... more The present paper continues the investigation of the relation between abstract ergodic theorems and fixed point theorems for affine and linear operators. The decomposition theorem for the ergodic subspace of a family of linear operators (cf. [1], [4], and [5]) is extended to families of affine operators. We use Dotson's generalization [3] of Eberlein's abstract ergodic theorem [4] to obtain
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
In this note we give an elementary proof that an arbitrary convex function can be uniformly appro... more In this note we give an elementary proof that an arbitrary convex function can be uniformly approximated by a convex C ∞-function on any closed bounded subinterval of the domain. An interesting byproduct of our proof is a global equation for a polygonal (piecewise affine) function.
Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asym... more Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.
The paper studies isolated spectral points of elements of Banach algebras and of bounded linear o... more The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that 0 is isolated in the spectrum of a bounded linear operator T if the (not necessarily closed) space M = {x : limn T n x 1/n = 0} is nonzero and complemented by a closed subspace N satisfying T N ⊂ N ⊂ T X.
We study decompositions of a Banach space operator T in the form T = K + Q, where K is compact (o... more We study decompositions of a Banach space operator T in the form T = K + Q, where K is compact (or meromorphic) and where the spectrum of Q is contained in the set of all accumulation points of the spectrum of T. Many known decomposition results of this type are subsumed in our construction. We prove that every Hilbert space operator has the meromorphic decomposition, and obtain an improvement of a result of Laurie and Radjavi on the West decomposition of Riesz operators in a Banach space.
In this paper we obtain a new inequality of Hilbert type for a finite number of nonnegative seque... more In this paper we obtain a new inequality of Hilbert type for a finite number of nonnegative sequences of real numbers from which we can recover as a special case an inequality due to Pachpatte. We also obtain an integral variant of the inequality.
In this paper we continue our investigation of multivariable integral inequalities of the type co... more In this paper we continue our investigation of multivariable integral inequalities of the type considered by Hilbert and recently by Pachpatte by focusing on fractional derivatives. Our results apply to integrable not necessarily continuous functions, and we are able to relax the original conditions to admit negative exponents in the weight functions.
A parameter dependent nonlinear differential-delay equation in a Banach space is investigated. It... more A parameter dependent nonlinear differential-delay equation in a Banach space is investigated. It is shown that if at the critical value of the parameter the problem satisfies a condition of linearized stability then the problem exhibits a stability which is uniform with respect to the whole range of the parameter values. The general theorem is applied to a diffusion system with applications in biology.
In this note we present an integral representation for the Drazin inverse A D of a complex square... more In this note we present an integral representation for the Drazin inverse A D of a complex square matrix A. This representation does not require any restriction on its eigenvalues.
The paper gives generalisations of Hardy's integral inequality that further extend results of Moh... more The paper gives generalisations of Hardy's integral inequality that further extend results of Mohapatra and Russell (Aequationes Math. 28 (1985), 199-207) and other authors by using α-submultiplicative or α-supermultiplicative functions.
For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ ... more For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ C such that T − λI is upper semi-B-Fredholm and ind (T − λI) ≤ 0, and let E a (T) be the set of all isolated eigenvalues of T in the approximate point spectrum σ a (T) of T. We say that T satisfies generalized a-Weyl's theorem if σ SBF − + (T) = σ a (T)\E a (T). Among other things, we show in this paper that if T satisfies generalized a-Weyl's theorem, then it also satisfies generalized Weyl's theorem σ BW (T) = σ(T) \ E(T), where σ BW (T) is the B-Weyl spectrum of T and E(T) is the set of all eigenvalues of T which are isolated in the spectrum of T. 2000 AMS subject classification: 47A53, 47A55
The paper introduces a special type of a Drazin-like inverse for closed linear operators that ari... more The paper introduces a special type of a Drazin-like inverse for closed linear operators that arises naturally in ergodic theory of operator semi-groups and cosine operator functions. The Drazin inverse for closed linear op-erators defined by Nashed and Zhao [30] and in a more general form by Koliha and Tran [21] is not sufficiently general to be applicable to operator semi-groups. The a-Drazin inverse is in general a closed, not necessarily bounded, operator. The paper gives applications of the inverse to partial differential equations.
Rendiconti del Circolo Matematico di Palermo, 1998
We study spectral sets of elements of a Banach algebra in terms of the existence of idempotent pa... more We study spectral sets of elements of a Banach algebra in terms of the existence of idempotent partitions of the unit of the algebra. Circularly isolated spectral sets are characterized, and a new Mbekhta decomposition for bounded linear operators is obtained.
The present paper continues the investigation of the relation between abstract ergodic theorems a... more The present paper continues the investigation of the relation between abstract ergodic theorems and fixed point theorems for affine and linear operators. The decomposition theorem for the ergodic subspace of a family of linear operators (cf. [1], [4], and [5]) is extended to families of affine operators. We use Dotson's generalization [3] of Eberlein's abstract ergodic theorem [4] to obtain
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