In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm ineq...In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.展开更多
Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k...Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operator展开更多
文摘In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.
基金Supported by the National Natural Science Foundation of China （10871224）
文摘Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operator
