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Papers by Kresimir Veselic

Research paper thumbnail of Perturbation of eigenvalues of the Klein-Gordon operators

We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds fo... more We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study operators of the form JG, where J, G are selfadjoint operators on a Hilbert space, J = J^* = J^-1 and G is positive definite and then we apply these results to obtain bounds of the Klein-Gordon eigenvalues under the change of the electrostatic potential. The developed general theory allows applications to some other instances, as e.g. the Sturm-Liouville problems with indefinite weight.

Research paper thumbnail of On weakly formulated Sylvester equations and applications

We use a "weakly formulated" Sylvester equation A^1/2TM^-1/2-A^-1/2TM^1/2=F to obtain n... more We use a "weakly formulated" Sylvester equation A^1/2TM^-1/2-A^-1/2TM^1/2=F to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. Another application is a bound for the square root of a positive selfadjoint operator which extends the known rule: "The relative error in the square root is bounded by the one half of the relative error in the radicand". Both bounds are illustrated on differential operators which are defined via quadratic forms.

Research paper thumbnail of The Tan 2 Θ-Theorem in Fluid Dynamics

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is c... more We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.

Research paper thumbnail of Bounds for contractive semigroups and second order systems

We derive a uniform bound for the difference of two contractive semigroups, if the difference of ... more We derive a uniform bound for the difference of two contractive semigroups, if the difference of their generators is form-bounded by the Hermitian parts of the generators themselves. We construct a semigroup dynamics for second order systems with fairly general operator coefficients and apply our bound to the perturbation of the damping term. The result is illustrated on a dissipative wave equation. As a consequence the exponential decay of some second order systems is proved.

Research paper thumbnail of Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded qu... more The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

Research paper thumbnail of Catenary — an elementary and complete approach

Dedicated to Professor Svetozar Kurepa on the occasion of his eightieth birthday The equilibrium ... more Dedicated to Professor Svetozar Kurepa on the occasion of his eightieth birthday The equilibrium of a standard catenary is solved without previous knowledge of the Variational Calculus. An elementary proof of the strict global minimum is provided. The equilibrium of the catenary is serving as a beautiful example of constrained minimisation. We have to minimise the functional x1 Ψ(y) = ρg y 0 √ 1 + y ′2dx, (1) where ρ (mass density) and g (acceleration of gravity) are positive constants, The minimisation is made among all continuously differentiable functions y satisfying the constraint x1 √ 1 + y ′2dx = d. (2) We pose the boundary conditions as either (both ends fixed) or (the right end sliding along the line y = −αx). 0 y(0) = 0, y(x1) = y1 (3) y(0) = 0, y(x1) = −αx1, α> 0 (4) After introducing the fundamentals of the classical variational calculus the extremal hyperbolic cosine is readily found, at least for the boundary conditions (3); the boundary conditions (4) need more the...

Research paper thumbnail of Diagonalization of indefinite saddle point forms

arXiv: Mathematical Physics, 2017

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-... more We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes Operator from fluid dynamics.

Research paper thumbnail of Protecting points from operator pencils

arXiv: Spectral Theory, 2019

We classify all sets of the form bigcuptinmathbbRmathrmspec(A+tB)\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)bigcuptinmathbbRmathrmspec(A+tB) where AAA and BBB ... more We classify all sets of the form bigcuptinmathbbRmathrmspec(A+tB)\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)bigcuptinmathbbRmathrmspec(A+tB) where AAA and BBB are self-adjoint operators and BBB is bounded, non-negative, and non-zero. We show that these sets are exactly the complements of those subsets of mathbbR\mathbb{R}mathbbR which are at most countable and contain none of their accumulation points.

Research paper thumbnail of Approximation of Green's Function and Application

Glasnik Matematicki, 2000

We consider a non local boundary value problem for el- liptic operator on a two dimensional domai... more We consider a non local boundary value problem for el- liptic operator on a two dimensional domain with a small hole around origin. The precise asymptotics in terms of diameter of the hole of values of solution on boundary of the hole is described by appropriate values of the Green function associated with the origin. In case of elliptical hole it is proved that solutions converge uniformly toward the Green function associated with the origin as diameter of the ellipse tends to zero.

Research paper thumbnail of On spectral condition of J-Herminian operators

Glasnik Matematicki, 2000

The spectral condition of a matrix H is the inm um of the condition numbers (Z) = kZkkZ 1 k, take... more The spectral condition of a matrix H is the inm um of the condition numbers (Z) = kZkkZ 1 k, taken over all Z such that Z 1 HZ is diagonal. This number controls the sensitivity of the spectrum of H under perturbations. A matrix is called J-Hermitian if H = JHJ for some J = J = J 1 . When diagonalizing J-Hermitian matrices it is natural to look at J-unitary Z, that is, those that satisfy Z JZ = J. Our rst result is: if there is such J-unitary Z, then the inm um above is taken on J-unitary Z, that is, the J unitary diagonalization is the most stable of all. For the special case when J-Hermitian matrix has denite spectrum, we give various upper bounds for the spectral condition, and show that all J-unitaries Z which diagonalize such a matrix have the same condition number. Our estimates are given in the spectral norm and the Hilbert{Schmidt norm. Our results are, in fact, formulated and proved in a general Hilbert space (under an appropriate generalization of the notion of 'diagona...

Research paper thumbnail of Passive Control of Linear Systems

We propose a method for designing optimal damping viscosities of dampers in order to calm down vi... more We propose a method for designing optimal damping viscosities of dampers in order to calm down vibrations of a structure with given mass and stiffness parameters. Our method is based on the minimization of the trace of the Lyapunov equation in the underlying phase space equipped with the energy norm. We compare our method with other common approaches. AMS subject classification: 70J25, 70Q05

Research paper thumbnail of Perturbation of eigenvalues of the Klein–Gordon operators

Revista Matemática Complutense, 2019

Research paper thumbnail of Index in semisimple Banach algebras

Glasnik Matematicki, 1982

Research paper thumbnail of A note on the accuracy of symmetric eigenreduction algorithms

Electronic Transactions on Numerical Analysis Etna, 1996

Research paper thumbnail of Special issue: Proceedings of the international workshop on accurate solution of eigenvalue problems. Pennsylvania State Univ., State College, PA, USA, July 20–23, 1998

Linear Algebra and Its Applications, 2000

Research paper thumbnail of A relative perturbation bound for positive definite matrices

Linear Algebra and its Applications, 1998

Research paper thumbnail of International Workshop on the Accurate Solution of Eigenvalue Problems (IWASEP IV)

Linear Algebra and Its Applications, 2006

Research paper thumbnail of New fast and accurate Jacobi SVD algorithm

Abstract. This paper presents new,implementation,of one{sided Jacobi SVD for triangular matrices ... more Abstract. This paper presents new,implementation,of one{sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors. New pivot strategy exploits the triangular form and uses the fact that the input triangular matrix is the result of rank revealing QR factorization. If used in the preconditioned Jacobi

Research paper thumbnail of A spectral theorem for a class of J-normal operators

Research paper thumbnail of On the perturbation theory for a class of J-selfadjoint operators

Research paper thumbnail of Perturbation of eigenvalues of the Klein-Gordon operators

We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds fo... more We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study operators of the form JG, where J, G are selfadjoint operators on a Hilbert space, J = J^* = J^-1 and G is positive definite and then we apply these results to obtain bounds of the Klein-Gordon eigenvalues under the change of the electrostatic potential. The developed general theory allows applications to some other instances, as e.g. the Sturm-Liouville problems with indefinite weight.

Research paper thumbnail of On weakly formulated Sylvester equations and applications

We use a "weakly formulated" Sylvester equation A^1/2TM^-1/2-A^-1/2TM^1/2=F to obtain n... more We use a "weakly formulated" Sylvester equation A^1/2TM^-1/2-A^-1/2TM^1/2=F to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. Another application is a bound for the square root of a positive selfadjoint operator which extends the known rule: "The relative error in the square root is bounded by the one half of the relative error in the radicand". Both bounds are illustrated on differential operators which are defined via quadratic forms.

Research paper thumbnail of The Tan 2 Θ-Theorem in Fluid Dynamics

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is c... more We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.

Research paper thumbnail of Bounds for contractive semigroups and second order systems

We derive a uniform bound for the difference of two contractive semigroups, if the difference of ... more We derive a uniform bound for the difference of two contractive semigroups, if the difference of their generators is form-bounded by the Hermitian parts of the generators themselves. We construct a semigroup dynamics for second order systems with fairly general operator coefficients and apply our bound to the perturbation of the damping term. The result is illustrated on a dissipative wave equation. As a consequence the exponential decay of some second order systems is proved.

Research paper thumbnail of Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded qu... more The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

Research paper thumbnail of Catenary — an elementary and complete approach

Dedicated to Professor Svetozar Kurepa on the occasion of his eightieth birthday The equilibrium ... more Dedicated to Professor Svetozar Kurepa on the occasion of his eightieth birthday The equilibrium of a standard catenary is solved without previous knowledge of the Variational Calculus. An elementary proof of the strict global minimum is provided. The equilibrium of the catenary is serving as a beautiful example of constrained minimisation. We have to minimise the functional x1 Ψ(y) = ρg y 0 √ 1 + y ′2dx, (1) where ρ (mass density) and g (acceleration of gravity) are positive constants, The minimisation is made among all continuously differentiable functions y satisfying the constraint x1 √ 1 + y ′2dx = d. (2) We pose the boundary conditions as either (both ends fixed) or (the right end sliding along the line y = −αx). 0 y(0) = 0, y(x1) = y1 (3) y(0) = 0, y(x1) = −αx1, α> 0 (4) After introducing the fundamentals of the classical variational calculus the extremal hyperbolic cosine is readily found, at least for the boundary conditions (3); the boundary conditions (4) need more the...

Research paper thumbnail of Diagonalization of indefinite saddle point forms

arXiv: Mathematical Physics, 2017

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-... more We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes Operator from fluid dynamics.

Research paper thumbnail of Protecting points from operator pencils

arXiv: Spectral Theory, 2019

We classify all sets of the form bigcuptinmathbbRmathrmspec(A+tB)\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)bigcuptinmathbbRmathrmspec(A+tB) where AAA and BBB ... more We classify all sets of the form bigcuptinmathbbRmathrmspec(A+tB)\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)bigcuptinmathbbRmathrmspec(A+tB) where AAA and BBB are self-adjoint operators and BBB is bounded, non-negative, and non-zero. We show that these sets are exactly the complements of those subsets of mathbbR\mathbb{R}mathbbR which are at most countable and contain none of their accumulation points.

Research paper thumbnail of Approximation of Green's Function and Application

Glasnik Matematicki, 2000

We consider a non local boundary value problem for el- liptic operator on a two dimensional domai... more We consider a non local boundary value problem for el- liptic operator on a two dimensional domain with a small hole around origin. The precise asymptotics in terms of diameter of the hole of values of solution on boundary of the hole is described by appropriate values of the Green function associated with the origin. In case of elliptical hole it is proved that solutions converge uniformly toward the Green function associated with the origin as diameter of the ellipse tends to zero.

Research paper thumbnail of On spectral condition of J-Herminian operators

Glasnik Matematicki, 2000

The spectral condition of a matrix H is the inm um of the condition numbers (Z) = kZkkZ 1 k, take... more The spectral condition of a matrix H is the inm um of the condition numbers (Z) = kZkkZ 1 k, taken over all Z such that Z 1 HZ is diagonal. This number controls the sensitivity of the spectrum of H under perturbations. A matrix is called J-Hermitian if H = JHJ for some J = J = J 1 . When diagonalizing J-Hermitian matrices it is natural to look at J-unitary Z, that is, those that satisfy Z JZ = J. Our rst result is: if there is such J-unitary Z, then the inm um above is taken on J-unitary Z, that is, the J unitary diagonalization is the most stable of all. For the special case when J-Hermitian matrix has denite spectrum, we give various upper bounds for the spectral condition, and show that all J-unitaries Z which diagonalize such a matrix have the same condition number. Our estimates are given in the spectral norm and the Hilbert{Schmidt norm. Our results are, in fact, formulated and proved in a general Hilbert space (under an appropriate generalization of the notion of 'diagona...

Research paper thumbnail of Passive Control of Linear Systems

We propose a method for designing optimal damping viscosities of dampers in order to calm down vi... more We propose a method for designing optimal damping viscosities of dampers in order to calm down vibrations of a structure with given mass and stiffness parameters. Our method is based on the minimization of the trace of the Lyapunov equation in the underlying phase space equipped with the energy norm. We compare our method with other common approaches. AMS subject classification: 70J25, 70Q05

Research paper thumbnail of Perturbation of eigenvalues of the Klein–Gordon operators

Revista Matemática Complutense, 2019

Research paper thumbnail of Index in semisimple Banach algebras

Glasnik Matematicki, 1982

Research paper thumbnail of A note on the accuracy of symmetric eigenreduction algorithms

Electronic Transactions on Numerical Analysis Etna, 1996

Research paper thumbnail of Special issue: Proceedings of the international workshop on accurate solution of eigenvalue problems. Pennsylvania State Univ., State College, PA, USA, July 20–23, 1998

Linear Algebra and Its Applications, 2000

Research paper thumbnail of A relative perturbation bound for positive definite matrices

Linear Algebra and its Applications, 1998

Research paper thumbnail of International Workshop on the Accurate Solution of Eigenvalue Problems (IWASEP IV)

Linear Algebra and Its Applications, 2006

Research paper thumbnail of New fast and accurate Jacobi SVD algorithm

Abstract. This paper presents new,implementation,of one{sided Jacobi SVD for triangular matrices ... more Abstract. This paper presents new,implementation,of one{sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors. New pivot strategy exploits the triangular form and uses the fact that the input triangular matrix is the result of rank revealing QR factorization. If used in the preconditioned Jacobi

Research paper thumbnail of A spectral theorem for a class of J-normal operators

Research paper thumbnail of On the perturbation theory for a class of J-selfadjoint operators