A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form [ aI cT dT ∗ bI ] . It reduces to a quadratic operator if d = 0. In this paper, spectra, norms, and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a...

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.–D. Choi et al., Rep. Math. Phys. 58, 77 (2006); Lin. Alg. Appl. 418, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to o...

For any n×n matrix A , we use the joint higher rank numerical range, Λk(A, . . . ,Am) , to define the higher rank numerical hull of A . We characterize the higher rank numerical hulls of Hermitian matrices. Also, the higher rank numerical hulls of unitary matrices are studied. Mathematics subject classification (2010): 15A60,81P68.

We study rank 2 bundles E on a two dimensional neighborhood of an irreducible curve C ≃ P1 with C2 = −k. Section 1 calculates bounds on the numerical invariants of E. Section 2 describes “balancing”, and proves the existence of families of bundles with prescribed numerical invariants. Section 3 studies rank 2 bundles on OP1(−k), giving an explicit construction of their moduli as stratified spaces.

‎let $v$ be an $n$-dimensional complex inner product space‎. ‎suppose‎ ‎$h$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎hrightarrow mathbb{c} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎denote by $v_{chi}(h)$ the symmetry class‎ ‎of tensors associated with $h$ and $chi$‎. ‎let $k(t)in‎ (v_{chi}(h))$ be the operator induced by $tin‎ ‎text{end}(v)$‎. ‎the...

in this paper, we introduce the notions of c-numerical range and c-spectrum of matrix polynomials. some algebraic and geometrical properties are investigated. we also study the relationship between the c-numerical range of a matrix polynomial and the joint c-numerical range of its coefficients.

For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert ...

An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed Subspace-Orbit Randomized singular value decomposition (SORSVD), which makes use of random sampling techniques to give an approximation to a low-rank matrix....