In this paper, we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set
E
. Starting from a frame
(
x
n
)
n
=
1
...∞
and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of
(
x
n
)
n
∈
E
c
so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of
(
x
n
)
n
=
1
∞
, which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame.
Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which ...is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.
Let X be a right Hilbert module over a C⁎-algebra A equipped with the canonical operator space structure. We define an elementary operator on X as a map ϕ:X→X for which there exists a finite number ...of elements ui in the C⁎-algebra B(X) of adjointable operators on X and vi in the multiplier algebra M(A) of A such that ϕ(x)=∑iuixvi for x∈X. If X=A this notion agrees with the standard notion of an elementary operator on A. In this paper we extend Mathieu's theorem for elementary operators on prime C⁎-algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert A-module X agrees with the Haagerup norm of its corresponding tensor in B(X)⊗M(A) if and only if A is a prime C⁎-algebra.
Let V be a countably generated Hilbert C^*-module over a C^*-algebra A. We prove that a sequence \{f_i:i\in I\}\subseteq V is a standard frame for V if and only if the sum \sum_{i\in I}\langle ...x,f_i\rangle\langle f_i,x\rangle converges in norm for every x\in V and if there are constants C,D>0 such that C\Vert x\Vert^2\le \Vert \sum_{i\in I}\langle x,f_i\rangle\langle f_i,x\rangle \Vert \le D\Vert x\Vert^2 for every x\in V. We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert C^*-modules over the C^*-algebra of all compact operators on some Hilbert space is discussed.
In this paper, we discuss a version of Birkhoff-James orthogonality in the
-algebra of all bounded linear operators on a finite-dimensional Hilbert space. We obtain a characterization of this ...relation and use it to describe some classes of operators. We also characterize linear preservers of this type of orthogonality.
In this paper we characterize the Birkhoff–James orthogonality for elements of a Hilbert C∗-module in terms of states of the underlying C∗-algebra. We also show that the Birkhoff–James orthogonality ...in a Hilbert C∗-module over a C∗-algebra A and orthogonality with respect to the A-valued inner product coincide if and only if A is isomorphic to C. In addition, some new results concerning the case of equality in the triangle inequality for elements of a Hilbert C∗-module are obtained.
We characterize the class of surjective (conjugate) linear mappings Φ:B(H)→B(H) that preserve the strong Birkhoff–James orthogonality in both directions. We also give characterizations of rank-one ...operators as well as coisometries in terms of the strong Birkhoff–James orthogonality.
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