For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a clas...For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.展开更多
For a compact operator tuple A, if its projective spectrum P(A*) is smooth, then there exists a natural Hermitian holomorphic line bundle EAover P(A*) which is a unitary invariant for A. This paper shows that under so...For a compact operator tuple A, if its projective spectrum P(A*) is smooth, then there exists a natural Hermitian holomorphic line bundle EAover P(A*) which is a unitary invariant for A. This paper shows that under some additional spectral conditions, EAis a complete unitary invariant, i.e., EAcan determine the compact operator tuple up to unitary equivalence.展开更多
This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H^2(D^2). A closed subspace M in H^2(D^2) is ...This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H^2(D^2). A closed subspace M in H^2(D^2) is called a submodule if ziM ? M(i = 1, 2). An associated integral operator(defect operator) CM captures much information about M. Using a Kre??n space indefinite metric on the range of CM, this paper gives a representation of M. Then it studies the group(called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup(called little Lorentz group) which turns out to be a finer invariant for M.展开更多
文摘For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.
基金supported by National Natural Science Foundation of China(Grant No.11671078)。
文摘For a compact operator tuple A, if its projective spectrum P(A*) is smooth, then there exists a natural Hermitian holomorphic line bundle EAover P(A*) which is a unitary invariant for A. This paper shows that under some additional spectral conditions, EAis a complete unitary invariant, i.e., EAcan determine the compact operator tuple up to unitary equivalence.
基金supported by Grant-in-Aid for Young Scientists(B)(Grant No.23740106)
文摘This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H^2(D^2). A closed subspace M in H^2(D^2) is called a submodule if ziM ? M(i = 1, 2). An associated integral operator(defect operator) CM captures much information about M. Using a Kre??n space indefinite metric on the range of CM, this paper gives a representation of M. Then it studies the group(called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup(called little Lorentz group) which turns out to be a finer invariant for M.
