Partial Isometries Research Articles

A note on completion to the unitary matrices

ABSTRACT Let be a complex matrix of order n. In this paper, once a square submatrix is fixed, we present several necessary conditions on , , and , where U is a unitary matrix. Particularly, given a unitary matrix , we characterize the completions and to some unitary matrix, whenever and are partial isometries.

A combinatorial proof of the extension property for partial isometries

A combinatorial proof of the extension property for partial isometries

On Algebras Generated by a Partial Isometry

In this paper we describe when a partial isometry generates an amenable $$C^*$$ -algebra or an amenable Banach algebra.

Normal elements on the generalized moore penrose inverse

In a ring with involution we already know that the Moore Penrose inverse can be used to construct the properties of normal elements. By using the fact that the group inverse of an element in a ring will be commutative with element which is commutative with that element, this paper explains that the generalization of Moore Penrose inverse can be also to establish some properties of normal elements in a ring with an involution. Some elements in the ring such as symmetric, EP, partial isometries, etc are also can be expressed in group inverse. So the results of this paper much be required for built the properties of those elements.

Partial isometries and a general spectral theorem

‎We prove a general spectral theorem for an arbitrary densely defined closed linear operator $T$ between complex Hilbert‎ ‎spaces $H$ and $K$‎. ‎The corresponding operator measure is partial isometry valued and has properties similar to those of the resolution of‎ ‎the identity of a non-negative self-adjoint operator‎. ‎The main method is the use of the canonical factorization (polar decomposition) obtained‎ ‎by v‎. ‎Neumann and Murray‎. ‎The uniqueness of the generalized resolution of the identity is studied together with the properties of a (non-multiplicative)‎ ‎functional calculus‎. ‎The properties of this generalized resolution of the identity are also investigated‎.

A remark on the partial isometry associated to the generalized polar decomposition of a matrix

Let T,M∈Cm×n be such that rank(M)=rank(T). Let UT and UM be the partial isometries associated to the generalized polar decompositions T=UT|T| and M=UM|M|. A formula for UM is provided in the Hermitian case, and based on this formula, some necessary and sufficient conditions are derived under which UM=UT.

On phaseless compressed sensing with partially known support

We establish a theoretical framework for the problem of phaseless compressed sensing with partially known signal support, which aims at generalizing the Null Space Property and the Strong Restricted Isometry Property from phase retrieval to partially sparse phase retrieval. We first introduce the concepts of the Partial Null Space Property (P-NSP) and the Partial Strong Restricted Isometry Property (P-SRIP); and then show that both the P-NSP and the P-SRIP are exact recovery conditions for the problem of partially sparse phase retrieval. We also prove that a random Gaussian matrix $ A\in \mathbb{R}^{m\times n} $ satisfies the P-SRIP with high probability when $ m = O(t(k-r)\log(\frac{n-r}{t(k-r)})). $

Some characterizations of partial isometry elements in rings with involutions

We give some sufficient and necessary conditions for an element in a ring with involution to be a partial isometry by using certain equations admitting solutions in a definite set.

WEAK NORMAL PROPERTIES OF PARTIAL ISOMETRIES

WEAK NORMAL PROPERTIES OF PARTIAL ISOMETRIES

Extending partial isometries of generalized metric spaces

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.

On the ranks of certain semigroups of order-preserving partial isometries of a finite chain

Let $X_n={1,2,ldots,n}$ be a finite chain, $mathcal{ODP}_{n}$ be the semigroup of order-preserving partial isometries on $X_n$ and $N$ be the set of all nilpotents in $mathcal{ODP}_{n}$. In this work, we study the nilpotents in $mathcal{ODP}_{n}$ and investigate the ranks of two subsemigroups of $mathcal{ODP}_{n}$; the nilpotent generatedsubsemigroup $langle Nrangle$ and the subsemigroup ~$L(n,r)= { alpha in mathcal{ODP}_{n} : |im~alpha|leq r}$.

Geometry of Interaction for MALL via Hughes--Van Glabbeek Proof-Nets

This article presents, for the first time, a Geometry of Interaction (GoI) interpretation inspired from Hughes--Van Glabbeek (HvG) proof-nets for multiplicative additive linear logic (MALL). Our GoI dynamically captures HvG’s geometric correctness criterion—the toggling cycle condition—in terms of algebraic operators. Our new ingredient is a scalar extension of the *-algebra in Girard’s *-ring of partial isometries over a Boolean polynomial ring with literals of eigenweights as indeterminates. To capture feedback arising from cuts, we construct a finer-grained execution formula. The expansion of this execution formula is longer than that for collections of slices for multiplicative GoI, hence it is harder to prove termination. Our GoI gives a dynamical, semantical account of Boolean valuations (in particular, pruning sub-proofs), conversion of weights (in particular, α-conversion), and additive (co)contraction, peculiar to additive proof-theory. Termination of our execution formula is shown to correspond to HvG’s toggling criterion. The slice-wise restriction of our execution formula (by collapsing the Boolean structure) yields the well-known correspondence, explicit or implicit in previous works on multiplicative GoI, between the convergence of execution formulas and acyclicity of proof-nets. Feedback arising from the execution formula by restricting to the Boolean polynomial structure yields autonomous definability of eigenweights among cuts from the rest of the eigenweights.

The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators

Let $T$ be an adjointable operator between two Hilbert $C^*$-modules and $T^*$ be the adjoint operator of $T$. The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac12$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$, where $U$ is a partial isometry, $\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$, respectively. Based on this new characterization of the polar decomposition, an application to the study of centered operators is carried out.

Completely Contractive Extensions of Hilbert Modules Over Tensor Algebras

This paper studies completely contractive extensions of Hilbert modules over tensor algebras over $$C^*$$ -correspondences. Using a result of Sz-Nagy and Foias on triangular contractions, extensions are parametrized in terms of contractive intertwining maps between certain defect spaces. These maps have a simple description when initial data consists of partial isometries. Sufficient conditions for the vanishing and nonvanishing of completely contractive Hilbert module $${\text {Ext}}^1$$ are given that parallel results for the classical disc algebra.

The ranks of certain semigroups of partial isometries

Let $$I_{n}$$ be the symmetric inverse semigroup on $$X_{n}=\{1,\ldots ,n\}$$ , and let $$DP_{n}$$ and $$ODP_{n}$$ be its subsemigroups of partial isometries and of order-preserving partial isometries on $$X_{n}$$ under its natural order, respectively. In this paper we find the ranks of the subsemigroups $$DP_{n,r}=\{ \alpha \in DP_{n}:|\mathrm {im\, }(\alpha )|\le r\}$$ and $$ODP_{n,r}=\{ \alpha \in ODP_{n}: |\mathrm {im\, }(\alpha )|\le r\}$$ for $$2\le r\le n-1$$ .

C*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ϕ on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l2(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.

C*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ϕ on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l2(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.

THE SEMIGROUP OF PARTIAL CO-FINITE ISOMETRIES OF POSITIVE INTEGERS

The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is studied. We describe Green's relations on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$, its band and proved that $\mathbf{I}\mathbb{N}_{\infty}$ is a simple $E$-unitary $F$-inverse semigroup. We described the least group congruence $\mathfrak{C}_{\mathbf{mg}}$ on $\mathbf{I}\mathbb{N}_{\infty}$ and proved that the quotient-semigroup $\mathbf{I}\mathbb{N}_{\infty}/\mathfrak{C}_{\mathbf{mg}}$ is isomorphic to the additive group of integers. An example of a non-group congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is presented. Also we proved that a congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence if and only if its restriction onto an isomorphic copy of the bicyclic semigroup in $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence.

A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition

We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm compu...

A note on roots and powers of partial isometries

Let T be a bounded operator and let \(k \ge 2\) be an integer. We study in this paper the following question: T is a partial isometry implies that \(T^k\) is a partial isometry and conversely.