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Bounded linear operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
Exercise 5.7.16 (Partial isometries). Let H and K be Hilbert spaces. A partial isometry from H to K is an operator V ∈ B(H, K), such that the restriction V|(ker V)⊥ of V to the orthogonal complement of the kernel is an isometry. The space (ker V)⊥ is called the initial space of V, and Im V is called the final space of V. Prove that if V is a partial isometry, then Im V is closed. Prove also that V is a partial isometry if and only if one of the following conditions holds: V*V is an orthogonal projection.V* is a partial isometry.VV*V = V.
Operator Theory
Published in Hugo D. Junghenn, Principles of Analysis, 2018
If also T=V|T| $ T = V|T| $ , where V is a partial isometry and kerV=kerT $ \ker V = \ker T $ , then V=U $ V = U $ on ran|T| $ \mathrm{ran}\,|T| $ and V=0=U $ V = 0 = U $ on ker|T|=(ran|T|)⊥ $ \ker |T| = (\mathrm{ran}\,|T|)^\perp $ , hence V=U $ V = U $ .
Extensions of Some Matrix Inequalities via Matrix Means
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Proof. Let X*A12=UP be the polar decomposition such that U is a partial isometry and P ∈ Pn. Then (X*AX12)α=(UP2U*)α=UP2αU*=UP⋅P2(α−1)PU*=X*A12|X*A12|2(α−1)A12X12=X*A12(A12XX*A12)α−1A12X.
Coupled supersymmetry and ladder structures beyond the harmonic oscillator
Published in Molecular Physics, 2018
Cameron L. Williams, Nikhil N. Pandya, Bernhard G. Bodmann, Donald J. Kouri
Since we have that It follows that and . The equivalences are warranted as the operators are defined on the same subspaces. From the polar decomposition for closed operators [23], the first equality guarantees that for some partial isometry U. Switching roles gives for some partial isometry V.
State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras
Published in Dynamical Systems, 2019
We construct from a bipartite graph Let . Define two kinds of edges such that for the partition the edge is defined such that for all . For an edge such that , the edge is defined such that . Then the set consists of such edges, that is We define transition matrices by setting Let us denote by the transition matrix of the graph . Then we have We set the matrix which is the transition matrix of the bipartite graph . Let be the Cuntz–Krieger algebra for the matrix . Let be the canonical generating partial isometries for the algebra assigned by the edges (21) in the bipartite graph . We define projections in by Since the graph is bipartite, as in (9), we know that We define a partial isometry in by We then have
A groupoid approach to C*-algebras associated with λ-graph systems and continuous orbit equivalence of subshifts
Published in Dynamical Systems, 2020
In what follows, a left-resolving λ-graph system is assumed to satisfy condition (I) and hence to be essentially free. Let be the normalizer of in consisting of partial isometries defined by For both elements and belong to . For satisfying we know that belongs to and satisfies under the natural identification between the algebras and We will consider the Weyl groupoid of germs of pairs with in the following way. Two elements with for i = 1, 2 are said to be equivalent if and there exists an open neighbourhood of such that for all . The set of equivalence classes of the pairs is denoted by with partially defined product: and the inverse operation The topology on is generated by It becomes an étale groupoid by a general theory studied by Renault ([22, Proposition 4.10], cf. [3, 23]). Since the groupoid is ample, for there exists a partial isometry such that . This fact was kindly informed by the referee. For a word let us denote by the length k. By [22, Proposition 4.13] proved by Renault, we have