Gradient Flows Computing the C-numerical Range with Applications in NMR Spectroscopy

In this paper gradient flows on unitary matrices are studied that maximize the real part of the C-numerical range of an arbitrary complex n×n-matrix A. The geometry of the C-numerical range can be quite complicated and is only partially understood. A numerical discretization scheme of the gradient flow is presented that converges to the set of critical points of the cost function. Special emphasis is taken on a situation arising in NMR spectroscopy where the matrices C,A are nilpotent and the C-numerical range is a circular disk in the complex plane around the origin.     

Operator norms,multiplicativity factors,and C-numerical radii

Let V be a normed vector space over C, let $\text{B}$(V) denote the algebra of linear bounded operators on V, and let N be an arbitrary seminorm or norm on $\text{B}$(V). In this paper we discuss multiplicativity factors for N, i.e., constants μ>0 for which N_μ≡μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C-numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.     

A new type of C -numerical range arising in quantum computing

We consider a new type of numerical range motivated by recent applications in quantum computing. We term the object of interest local C -numerical rangeW_loc(C, A) of A. It is obtained by replacing the special unitary group in the definition of the C -numerical range by the so-called local subgroup of SU (2^N ), i.e. by the N -fold tensor product SU (2) ⊗ · · · ⊗ SU(2) of unitary (2 × 2)-matrices. First, it is shown that the local C -numerical range has rather unusual geometric properties compared to the ordinary one, e.g. it is in general neither star-shaped nor simply connected. Then two numerical algorithms, a Newton and a conjugate gradient method on the Lie group SU (2) ⊗ · · · ⊗ SU (2), are demonstrated to maximize the real part of W_loc(C, A) which also gives a Euclidean measure of the so-called pure-state entanglement in quantum computing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conditions for the generalized numerical range to be real

Necessary and sufficient conditions are given for the C-numerical range of a matrix A to be a subset of the real axis. In particular, it is shown that both A and C must be translates of hermitian matrices.     

A condition for the convexity of the norm-numerical range of a matrix

Let v be a norm on Cⁿ, and let a be a matrix which has a v-Hermitian decomposition. (1) The v-numerical range of a is convex. (This generalizes the Hausdorff-Toeplitz theorem.) In fact, the v-numerical range is equal to the field of values of a matrix similar to a. (2) If the Hermitian and v-Hermitian decompositions of a coincide, then the v-numerical range of a and the field of values of a are the same. This follows from detailed information about the boundary of the range.     

The q-numerical radius of weighted shift operators with periodic weights

We deal with the q-numerical radius of weighted unilateral and bilateral shift operators. In particular, the q-numerical radius of weighted shift operators with periodic weights is discussed and computed.     

Norm properties of C-numerical radii

Given n×n complex matrices A, C, the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)≡ma{|tr(CU}{}^1\text{AU)|:U unitary}}$

. For C=diag(1,0,…,0) it reduces to the classical numerical radius $\text{r(A)= max{|x}{}^1\text{Ax|:x}{}^1\text{x=1}}$. We show that r_c is a generalized matrix norm if and only if C is nonscalar and trC≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v?0 for which vN is multiplicative. A technique to obtain such v is then applied to C-numerical radii with Hermitian C. In particular we find that vr is a matrix norm if and only if v?4.     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.     

Perturbation of the q-numerical radius of a weighted shift operator

We formulate the Taylor series expansion for the q-numerical radius of a weighted shift operator with periodic weights near q=0. Coefficients up to the fourth order in the expansion are found via the perturbation theory of Hermitian matrices.     

Spectral inclusion properties of the numerical range in a space with an indefinite metric

In this paper the numerical range of operators (possibly unbounded) in an indefinite inner product space is studied. In particular, we show that the spectrums of bounded positive operators (or the spectrum of unbounded uniformly I-positive operators) are contained in the closure of the I-numerical range.     

C-numerical ranges and C-numerical radii

In this paper, we give a brief survey on C-numerical ranges and C-numerical radii. New results are obtained and open problems are mentioned.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

Multiplicativity factors for C-numerical radii

For matrices $\text{A, C?}\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ , the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)=max{|tr(CU}{}^1\text{AU)|:Uunitary}}$

. This generalizes the classical numerical radius r(A). It is known that r_c constitutes a norm on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ if and only if C is nonscalar and trC≠0. For all such C we obtain multiplicativity factors for r_c, i.e., constant μ>0 for which μr_c is submultiplicative on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$.     

Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy

Incidence coalgebras C = K ^□ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K ^□ I is representation-directed if and only if the Euler quadratic form q _C : ?^(I) → ? of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K ^□ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K ^□ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K ^□ I that are right semiperfect.