Orthogonal polynomials of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" display="inline" overflow="scroll"><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>-linear generalized minimal residual method

Huhtanen, Marko; Perämäki, Allan

doi:10.1016/j.jat.2012.12.007

Cited by 4 publications

(11 citation statements)

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“…In what follows it is shown, by partly following [12], that there exists a correspondence between (discrete) biradial measures and antilinear Jacobi operators. For convenience, the following notation for the monomials appearing in Definition 2.1 is employed.…”

Section: Biradial Measures With Finite Support and Antilinear Jacobi ...mentioning

confidence: 88%

“…In what follows, we are more interested in the antilinear case, i.e., when C = 0. To this end, the following space of polynomials was introduced in [12].…”

Section: Functions Of Antilinear Operators and Spectral Theorymentioning

confidence: 99%

“…Of course, polynomials in complex linear operators appear regularly. Polynomials in antilinear operators have recently proved useful in numerical linear algebra and approximation theory [5,11,12]. A first notable difference between these two extremes is the fact that a polynomial in an antilinear operator typically becomes genuinely a real linear operator, i.e., it has nonzero complex and antilinear parts.…”

Section: Functions Of Antilinear Operators and Spectral Theorymentioning

confidence: 99%

“…In finite dimensional H, sufficiency is established by the notion of contriangularizability, which is equivalent to σ(A 2 ) ⊂ [0, ∞) [9]. (For the probability of being contriangularizable, see [12]. )…”

Section: Moreover If a Is Self-adjoint Then This Further Holds For Al...mentioning

confidence: 99%

“…In real linear operator theory, complex linear operators constitute one extreme while antilinear operators constitute the other one. There is strong evidence to expect that these two extremes turn out to be mathematically equally rich [12,13,10,11]. In this paper, a function space structure is identified which has an analogous connection with antilinear operators as complex analysis has with complex linear operators.…”

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confidence: 96%

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Function theory and spectral mapping theorems for antilinear operators

Huhtanen¹,

Perämäki²

2014

J. Operator Theory

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Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping theorems. These spectral mapping theorems are inclusive in general. Equality can be established in the self-adjoint case. The arising functions are shown to possess a biradial character. It is shown that to any given set of Jacobi parameters corresponds a biradial measure yielding these parameters in an iterative orthogonalization process in this function space, once equipped with the corresponding L 2 structure.

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Section: Biradial Measures With Finite Support and Antilinear Jacobi ...mentioning

confidence: 88%

“…In what follows, we are more interested in the antilinear case, i.e., when C = 0. To this end, the following space of polynomials was introduced in [12].…”

Section: Functions Of Antilinear Operators and Spectral Theorymentioning

confidence: 99%

Section: Functions Of Antilinear Operators and Spectral Theorymentioning

confidence: 99%

Section: Moreover If a Is Self-adjoint Then This Further Holds For Al...mentioning

confidence: 99%

mentioning

confidence: 96%

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Function theory and spectral mapping theorems for antilinear operators

Huhtanen¹,

Perämäki²

2014

J. Operator Theory

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Polynomials and lemniscates of indefiniteness

Huhtanen

Nevanlinna

2015

Numer. Math.

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For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in C n×n . A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the L 2 -norm algorithms for solving the problem are devised.

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Anti- (conjugate) linearity

Uhlmann

2016

Sci. China Phys. Mech. Astron.

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This is an introduction to antilinear operators. In following E. P. Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear".By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case.Basics of antilinearity is explained in sections 2, 3, 4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal operators, (skew) conjugations, involutions, and acq-lines, the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines. as well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, associated to a vector of a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces. 1 2

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Orthogonal polynomials of the R-linear generalized minimal residual method

Cited by 4 publications

References 18 publications

Function theory and spectral mapping theorems for antilinear operators

Function theory and spectral mapping theorems for antilinear operators

Polynomials and lemniscates of indefiniteness

Anti- (conjugate) linearity

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