Representations for generalized Drazin inverse of operator matrices over a Banach space

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature

Thermodynamic length in open quantum systems

The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.Comment: 22 pages, 3 figures. v5: minor corrections, accepted in Quantu

Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open non unitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples.Comment: 29 pages, 4 figures, expanded historical citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht

ADDITIVE PROPERTIES OF THE DRAZIN INVERSE FOR MATRICES AND BLOCK REPRESENTATIONS: A SURVEY

In this paper, a review of a development of the Drazin inverse for the sum of two matrices has been given. Since this topic is closely related to the problem of finding the Drazin inverse of a 2x2 block matrix, the paper also offers a survey of this subject

A characterization and representation of the generalized inverse A(2)T,S and its applications

AbstractThis paper presents an explicit expression for the generalized inverse A(2)T,S. Based on this, we established the characterization, the representation theorem and the limiting process for A(2)T,S. As an application, we estimate the error bound of the iterative method for approximating A(2)T,S

Some additive results on Drazin inverse

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aa^D(a+b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b)^D. As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.Supported by the National Natural Science Foundation of China (11361009), the Guangxi Provincial Natural Science Foundation of China (2013GXNSFAA019008), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities.Liu, X.; Qin, X.; Benítez López, J. (2015). Some additive results on Drazin inverse. Applied Mathematics - A Journal of Chinese Universities. 30(4):479-490. https://doi.org/10.1007/s11766-015-3333-4S479490304A Ben-Israel, T N E Greville. Generalized Inverses, Theory and Applications, 2nd edition, Springer-Verlag, 2003.S L Campbell, C D Meyer. Generalized Inverses of Linear Transformations, Pitman (Advanced Publishing Program), Boston, MA, 1979.N Castro-González, J J Koliha. Additive perturbation results for the Drazin inverse, Linear Algebra Appl, 2005, 397: 279–297.N Castro-González, E Dopazo, M F Martínez-Serrano. On the Drazin inverse of the sum of two operators and its application to operator matrices, J Math Anal Appl, 2008, 350: 207–215.N Castro-González, M F Martínez-Serrano. Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra, Linear Algebra Appl, 2010, 432: 1885–1895.N Castro-González, J J Koliha. New additive results for the Drazin inverse, Proc Roy Soc Edinburgh Sect A, 2004, 134: 1085–1097.M Catral, D D Olesky, P van den Driessche. Block representations of the Drazin inverse of a bipartite matrix, Electron J Linear Algebra, 2009, 18: 98–107.J L Chen, G F Zhuang, Y Wei. The Drazin inverse of a sum of morphisms, Acta Math Sci Ser A Chin Ed, 2009, 29(3): 538–552.D S Cvetković-Ilić, D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl, 2006, 418, 53–61.D S Cvetković-Ilić. A note on the representation for the Drazin inverse of 2 × 2 block matrices, Linear Algebra Appl, 2008, 429: 242–248.C Deng. The Drazin inverses of sum and difference of idempotents, Linear Algebra Appl, 2009, 430: 1282–1291.C Deng, Y Wei. Characterizations and representations of the Drazin inverse of idempotents, Linear Algebra Appl, 2009, 431: 1526–1538.C Deng, Y Wei. New additive results for the generalized Drazin inverse, J Math Anal Appl, 2010, 370: 313–321.D S Djordjević, P S Stanimirović. On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math J, 2001, 51(126): 617–634.D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse, J Aust Math Soc, 2002, 73: 115–125.D S Djordjević, V Rakočević. Lectures on Generalized inverses, University of Niš, 2008.E Dopazo, M F Martínez-Serrano. Further results on the representation of the Drazin inverse of a 2 × 2 block matrices, Linear Algebra Appl, 2010, 432: 1896–1904.M P Drazin. Pseudo-inverses in associative rings and semiproup, Amer Math Monthly, 1958, 65: 506–514.R E Hartwig, G R Wang, Y Wei. Some additive results on Drazin inverse, Linear Algebra Appl, 2001, 322: 207–217.R E Hartwig, X Li, Y Wei. Representations for the Drazin inverse of a 2×2 block matrix, SIAM J Matrix Anal Appl, 2006, 27: 757–771.Y Liu, C G Cao. Drazin inverse for some partitioned matrices over skew fields, J Nat Sci Heilongjiang Univ, 2004, 24: 112–114.J Ljubisavljević, D S Cvetković-Ilić. Additive results for the Drazin inverse of block matrices and applications, J Comput Appl Math, 2011, 235: 3683–3690.C D Meyer ffixJr, N J Rose. The index and the Drazin inverse of block triangular matrices, SIAM J Appl Math, 1977, 33(1): 1–7.L Wang, H H Zhu, X Zhu, J L Chen. Additive property of Drazin invertibility of elements, arXiv: 1307.1816v1 [math.RA], 2013.H Yang, X Liu. The Drazin inverse of the sum of two matrices and its applications, J Comput Appl Math, 2011, 235: 1412–1417