Name Palle Jorgensen. I am a Professor in a Mathematics department. And my time is divided between teaching and research, journal articles, and publication of books. Address: Iowa City, Iowa, United States
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded... more Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and operator theory of reflection positivity.
ABSTRACT We consider the role of finite-dimensional Lie algebras as symmetries of a class of simp... more ABSTRACT We consider the role of finite-dimensional Lie algebras as symmetries of a class of simple (i.e., no two-sided ideals) C * -algebras which arise by a quantum deformation of classical mechanics. The symmetries under consideration are defined relative to the Connes non-commutative differential geometric formalisms. Among the Lie algebras which occur, the Heisenberg-Lie algebra is shown to be generic. We then show how deformations of the Heisenberg generators lead to representations of the infinite C * -algebra defined on a Hilbert space (also called the Cuntz algebra). Finally, we consider the stability question for the deformations, where stability refers to the C * -algebraic isomorphism class.
Mathematicians love to find groups in unexpected places, and the discovery of Frobenius that the ... more Mathematicians love to find groups in unexpected places, and the discovery of Frobenius that the set of phases of the eigenvalues of maximal modulus of a given positive matrix is the finite union of finite abelian groups with multiplication equal to multiplication of phases is a case in point. This discovery has had all kinds of wonderful applications, for example to symbolic dynamics and computational mathematics. But the result does not carry over to the infinite-dimensional version of the Perron-Frobenius theory which you saw in the last chapter. So the next best thing is to look instead for a related property which still has the implications that you want for wavelet analysis. The group property that Frobenius discovered has its origin in the study of inner products u* υ where u and υ are eigenvectors for some given positive matrix R corresponding to a pair of peripheral eigenvalues, say λ u and λ v .
In the present paper, we consider the marginal entropy of the a pure state in the tensor product ... more In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton's result of asymptotical behavior of group integrals. For group in-tegrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces H A and H B. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2015
Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [F... more Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of tuples of nonnegative integers [Formula: see text] with [Formula: see text] for finitely many [Formula: see text], the [Formula: see text]-moment problem on [Formula: see text] entails determining whether or not there exists a measure [Formula: see text] on [Formula: see text] so that [Formula: see text] and [Formula: see text] We prove that [Formula: see text] exists if and only if a natural analogue of the Riesz–Haviland functional [Formula: see text] is [Formula: see text]-positive, i.e. if [Formula: see text] is any polynomial which is nonnegative for all [Formula: see text], then [Formula: see text] We will also provide a sufficient condition for [Formula: see text] to be unique, an analogue of a celebrated theorem of K. Schmüdgen and an application to stochastic processes.
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded... more Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and operator theory of reflection positivity.
ABSTRACT We consider the role of finite-dimensional Lie algebras as symmetries of a class of simp... more ABSTRACT We consider the role of finite-dimensional Lie algebras as symmetries of a class of simple (i.e., no two-sided ideals) C * -algebras which arise by a quantum deformation of classical mechanics. The symmetries under consideration are defined relative to the Connes non-commutative differential geometric formalisms. Among the Lie algebras which occur, the Heisenberg-Lie algebra is shown to be generic. We then show how deformations of the Heisenberg generators lead to representations of the infinite C * -algebra defined on a Hilbert space (also called the Cuntz algebra). Finally, we consider the stability question for the deformations, where stability refers to the C * -algebraic isomorphism class.
Mathematicians love to find groups in unexpected places, and the discovery of Frobenius that the ... more Mathematicians love to find groups in unexpected places, and the discovery of Frobenius that the set of phases of the eigenvalues of maximal modulus of a given positive matrix is the finite union of finite abelian groups with multiplication equal to multiplication of phases is a case in point. This discovery has had all kinds of wonderful applications, for example to symbolic dynamics and computational mathematics. But the result does not carry over to the infinite-dimensional version of the Perron-Frobenius theory which you saw in the last chapter. So the next best thing is to look instead for a related property which still has the implications that you want for wavelet analysis. The group property that Frobenius discovered has its origin in the study of inner products u* υ where u and υ are eigenvectors for some given positive matrix R corresponding to a pair of peripheral eigenvalues, say λ u and λ v .
In the present paper, we consider the marginal entropy of the a pure state in the tensor product ... more In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton's result of asymptotical behavior of group integrals. For group in-tegrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces H A and H B. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2015
Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [F... more Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of tuples of nonnegative integers [Formula: see text] with [Formula: see text] for finitely many [Formula: see text], the [Formula: see text]-moment problem on [Formula: see text] entails determining whether or not there exists a measure [Formula: see text] on [Formula: see text] so that [Formula: see text] and [Formula: see text] We prove that [Formula: see text] exists if and only if a natural analogue of the Riesz–Haviland functional [Formula: see text] is [Formula: see text]-positive, i.e. if [Formula: see text] is any polynomial which is nonnegative for all [Formula: see text], then [Formula: see text] We will also provide a sufficient condition for [Formula: see text] to be unique, an analogue of a celebrated theorem of K. Schmüdgen and an application to stochastic processes.
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