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Journal articles on the topic 'Numerical linear and multilinear algebra'

Author: Grafiati

Published: 18 May 2024

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1

Bini, Dario, Marilena Mitrouli, Marc Van Barel, and Joab Winkler. "Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications." Linear Algebra and its Applications 502 (August 2016): 1–4. http://dx.doi.org/10.1016/j.laa.2016.03.042.

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2

Huang, Zhengge, and Jingjing Cui. "Improved Brauer-type eigenvalue localization sets for tensors with their applications." Filomat 34, no. 14 (2020): 4607–25. http://dx.doi.org/10.2298/fil2014607h.

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In this paper, by excluding some sets from the Brauer-type eigenvalue inclusion sets for tensors developed by Bu et al. (Linear Algebra Appl. 512 (2017) 234-248) and Li et al. (Linear and Multilinear Algebra 64 (2016) 727-736), some improved Brauer-type eigenvalue localization sets for tensors are given, which are proved to be much tighter than those put forward by Bu et al. and Li et al. As applications, some new criteria for identifying the nonsingularity of tensors are developed, which are better than some previous results. This fact is illustrated by some numerical examples.

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3

Sahoo, Satyajit. "On A-numerical radius inequalities for 2 x 2 operator matrices-II." Filomat 35, no. 15 (2021): 5237–52. http://dx.doi.org/10.2298/fil2115237s.

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Rout et al. [Linear Multilinear Algebra 2020, DOI: 10.1080/03081087.2020.1810201] presented certain A-numerical radius inequalities for 2x2 operator matrices and further results on A-numerical radius of certain 2x2 operator matrices are obtained by Feki [Hacet. J. Math. Stat., 2020, DOI:10.15672/hujms.730574], very recently. The main goal of this article is to establish certain A-numerical radius equalities for operator matrices. Several new upper and lower bounds for the A-numerical radius of 2 x 2 operator matrices has been proved, where A be the 2 x 2 diagonal operator matrix whose diagonal entries are positive bounded operator A. Further, we prove some refinements of earlier A-numerical radius inequalities for operators.

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4

Khoromskij, B. N. "Structured Rank-(r1, . . . , rd) Decomposition of Function-related Tensors in R_D." Computational Methods in Applied Mathematics 6, no. 2 (2006): 194–220. http://dx.doi.org/10.2478/cmam-2006-0010.

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AbstractThe structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra shows an almost linear cost in the one-dimensional problem size. The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors.

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5

Benzi, Michele, and Ru Huang. "Some matrix properties preserved by generalized matrix functions." Special Matrices 7, no. 1 (January 8, 2019): 27–37. http://dx.doi.org/10.1515/spma-2019-0003.

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Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.

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6

Choi, Yun Sung, Domingo Garcia, Sung Guen Kim, and Manuel Maestre. "THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE." Proceedings of the Edinburgh Mathematical Society 49, no. 1 (February 2006): 39–52. http://dx.doi.org/10.1017/s0013091502000810.

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AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

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7

Qi, Liqun, Yimin Wei, Changqing Xu, and Tan Zhang. "Linear algebra and multilinear algebra." Frontiers of Mathematics in China 11, no. 3 (May 6, 2016): 509–10. http://dx.doi.org/10.1007/s11464-016-0540-0.

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8

Marcus, Marvin. "Multilinear methods in linear algebra." Linear Algebra and its Applications 150 (May 1991): 41–59. http://dx.doi.org/10.1016/0024-3795(91)90158-s.

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9

Qi, Liqun, Wenyu Sun, and Yiju Wang. "Numerical multilinear algebra and its applications." Frontiers of Mathematics in China 2, no. 4 (October 2007): 501–26. http://dx.doi.org/10.1007/s11464-007-0031-4.

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10

Gentle, James. "Matrix Analysis and Applied Linear Algebra, Numerical Linear Algebra, and Applied Numerical Linear Algebra." Journal of the American Statistical Association 96, no. 455 (September 2001): 1136–37. http://dx.doi.org/10.1198/jasa.2001.s412.

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11

Rota, Gian-Carlo. "Linear algebra and group representations, Vol. II, Multilinear algebra and group representations." Advances in Mathematics 57, no. 1 (July 1985): 91. http://dx.doi.org/10.1016/0001-8708(85)90107-0.

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12

Ning, Jing, Yajun Xie, and Jie Yao. "Efficient Splitting Methods for Solving Tensor Absolute Value Equation." Symmetry 14, no. 2 (February 15, 2022): 387. http://dx.doi.org/10.3390/sym14020387.

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The tensor absolute value equation is a class of interesting structured multilinear systems. In this article, from the perspective of pure numerical algebra, we first consider a tensor-type successive over-relaxation method (SOR) (called TSOR) and tensor-type accelerated over-relaxation method (AOR) (called TAOR) for solving tensor absolute value equations. Furthermore, one type of preconditioned tensor splitting method is also applied for solving structured multilinear systems. Numerical experiments adequately demonstrate the efficiency of the presented methods.

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13

Demmel, James W., Michael T. Heath, and Henk A. van der Vorst. "Parallel numerical linear algebra." Acta Numerica 2 (January 1993): 111–97. http://dx.doi.org/10.1017/s096249290000235x.

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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of paralled processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.

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14

Chen, Huanyin, and Marjan Abdolyousefi. "Generalized Jacobson’s lemma for generalized Drazin inverses." Filomat 35, no. 7 (2021): 2267–75. http://dx.doi.org/10.2298/fil2107267c.

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We present new generalized Jacobson?s lemma for generalized Drazin inverses. This extends the main results on g-Drazin inverse of Yan, Zeng and Zhu (Linear & Multilinear Algebra, 68(2020), 81-93).

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15

Stewart, G. W., Tom King, Yves Achdou, and Frank Stenger. "Book Review: Numerical linear algebra." Mathematics of Computation 68, no. 225 (January 1, 1999): 453–60. http://dx.doi.org/10.1090/s0025-5718-99-01069-8.

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16

Moradi, Hamid, Shigeru Furuichi, and Zahra Heydarbeygi. "New Refinement of the Operator Kantorovich Inequality." Mathematics 7, no. 2 (February 1, 2019): 139. http://dx.doi.org/10.3390/math7020139.

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We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].

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17

Moafian, F., and Ebrahimi Vishki. "Lie higher derivations on triangular algebras revisited." Filomat 30, no. 12 (2016): 3187–94. http://dx.doi.org/10.2298/fil1612187m.

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Motivated by the extensive works of W.-S. Cheung [Linear Multilinear Algebra, 51 (2003), 299-310] and X.F. Qi [Acta Math. Sinica, English Series, 29 (2013), 1007-1018], we present the structure of Lie higher derivations on a triangular algebra explicitly. We then study those conditions under which a Lie higher derivation on a triangular algebra is proper. Our approach provides a direct proof for some known results concerning to the properness of Lie higher derivations on triangular algebras.

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18

Chen, Huanyin, and Marjan Abdolyousefi. "New formulae of the Drazin inverse of anti-triangular complex block matrices." Filomat 36, no. 12 (2022): 4251–64. http://dx.doi.org/10.2298/fil2212251c.

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Let E, F ? Cnxn. If EFiE = 0 for all i ? N, we give the explicit representation of the Drazin inverse of the block complex matrix (E I F 0). We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear & Multilinear Algebra, 14(1983), 195-198].

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19

Chen, Huanyin, and Abdolyousefi Sheibani. "The group inverse of certain block complex matrices." Filomat 37, no. 10 (2023): 3153–66. http://dx.doi.org/10.2298/fil2310153c.

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We present new additive results for the group inverse of block complex matrices. As an application, the representations for the group inverse of a block complex matrix are given. These extend the main results of Ben?tez, Liu and Zhu (Linear Multilinear Algebra, 59(2011), 279-289).

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20

Ballard, Grey, James Demmel, Olga Holtz, and Oded Schwartz. "Minimizing Communication in Numerical Linear Algebra." SIAM Journal on Matrix Analysis and Applications 32, no. 3 (July 2011): 866–901. http://dx.doi.org/10.1137/090769156.

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21

Dongarra, Jack J., and Victor Eijkhout. "Numerical linear algebra algorithms and software." Journal of Computational and Applied Mathematics 123, no. 1-2 (November 2000): 489–514. http://dx.doi.org/10.1016/s0377-0427(00)00400-3.

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22

Eldén, Lars. "Numerical linear algebra in data mining." Acta Numerica 15 (May 2006): 327–84. http://dx.doi.org/10.1017/s0962492906240017.

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Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

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23

Kannan, Ravindran, and Santosh Vempala. "Randomized algorithms in numerical linear algebra." Acta Numerica 26 (May 1, 2017): 95–135. http://dx.doi.org/10.1017/s0962492917000058.

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This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.

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24

Bru, Rafael, and Juan Manuel Peña. "Preface: numerical and applied linear algebra." Advances in Computational Mathematics 35, no. 2-4 (July 6, 2011): 99–102. http://dx.doi.org/10.1007/s10444-011-9170-y.

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25

Kostlan, Eric. "Complexity theory of numerical linear algebra." Journal of Computational and Applied Mathematics 22, no. 2-3 (June 1988): 219–30. http://dx.doi.org/10.1016/0377-0427(88)90402-5.

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26

Arveson, W. "C*-Algebras and Numerical Linear Algebra." Journal of Functional Analysis 122, no. 2 (June 1994): 333–60. http://dx.doi.org/10.1006/jfan.1994.1072.

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27

Mastronardi, Nicola, and Sabine Van Huffel. "Numerical linear algebra and its applications." Numerical Linear Algebra with Applications 12, no. 8 (2005): 683. http://dx.doi.org/10.1002/nla.443.

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28

Gonzaga, Clovis Caesar, and Jin Yun Yuan. "Foz2006 numerical linear algebra and optimization." Numerical Linear Algebra with Applications 15, no. 10 (June 16, 2008): 887–89. http://dx.doi.org/10.1002/nla.601.

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29

Stroch, Joel A., and Christoph Börgers. "Introduction to Numerical Linear Algebra [Bookshelf]." IEEE Control Systems 44, no. 1 (February 2024): 79–80. http://dx.doi.org/10.1109/mcs.2023.3329927.

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30

Danchev, Peter, Esther García, and Miguel Gómez Lozano. "Decompositions of matrices into potent and square-zero matrices." International Journal of Algebra and Computation 32, no. 02 (January 31, 2022): 251–63. http://dx.doi.org/10.1142/s0218196722500126.

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In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

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31

Wang, Xiaoxiao, Chaoqian Li, and Yaotang Li. "A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices." Open Mathematics 16, no. 1 (April 2, 2018): 298–310. http://dx.doi.org/10.1515/math-2018-0030.

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AbstractA set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues different from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we fix n parameters in [0, 1] to give a new set including all eigenvalues different from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014) 74-87) and Li et al. (Linear and Multilinear Algebra 63(11) (2015) 2159-2170) for estimating the moduli of subdominant eigenvalues.

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32

Drygas, Hilmar. "Linear sufficiency and some applications in multilinear estimation." Journal of Multivariate Analysis 16, no. 1 (February 1985): 71–84. http://dx.doi.org/10.1016/0047-259x(85)90052-1.

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33

Choudhury, Projesh, Rajesh Kannan, and K. Sivakumar. "A note on linear preservers of semipositive and minimally semipositive matrices." Electronic Journal of Linear Algebra 34 (February 21, 2018): 687–94. http://dx.doi.org/10.13001/1081-3810.3864.

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Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--1862, 2016.] are clarified.

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34

Yang, Lei, Zheng-Hai Huang, and Yu-Fan Li. "A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery." Asia-Pacific Journal of Operational Research 32, no. 01 (February 2015): 1540008. http://dx.doi.org/10.1142/s0217595915400084.

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This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low multilinear-rank tensor recovery problem. The variable splitting technique and convex relaxation technique are used to transform this problem into a tractable constrained optimization problem. Considering the favorable structure of the problem, we develop a splitting augmented Lagrangian method (SALM) to solve the resulting problem. The proposed algorithm is easily implemented and its convergence can be proved under some conditions. Some preliminary numerical results on randomly generated and real completion problems show that the proposed algorithm is very effective and robust for tackling the low multilinear-rank tensor completion problem.

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35

Andersson, Mats, Oleg Burdakov, Hans Knutsson, and Spartak Zikrin. "Global Search Strategies for Solving Multilinear Least-Squares Problems." Sultan Qaboos University Journal for Science [SQUJS] 16 (April 1, 2012): 12. http://dx.doi.org/10.24200/squjs.vol17iss1pp12-21.

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The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by the results of numerical experiments performed for some problems related to the design of filter networks.

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36

Heo, Jaeseong, and Maria Joiţa. "Corrigendum to ‘A Stinespring type theorem for completely positive multilinear maps on Hilbert C*-modules’ [Linear Multilinear Algebra 67 (2019), 121–140]." Linear and Multilinear Algebra 67, no. 8 (May 23, 2019): 1715–16. http://dx.doi.org/10.1080/03081087.2019.1607818.

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37

Yoshikawa, Kohei, and Shuichi Kawano. "Multilinear Common Component Analysis via Kronecker Product Representation." Neural Computation 33, no. 10 (September 16, 2021): 2853–80. http://dx.doi.org/10.1162/neco_a_01425.

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Abstract We consider the problem of extracting a common structure from multiple tensor data sets. For this purpose, we propose multilinear common component analysis (MCCA) based on Kronecker products of mode-wise covariance matrices. MCCA constructs a common basis represented by linear combinations of the original variables that lose little information of the multiple tensor data sets. We also develop an estimation algorithm for MCCA that guarantees mode-wise global convergence. Numerical studies are conducted to show the effectiveness of MCCA.

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38

Appa, Gautam, Allan Findlay, Philip E. Gill, Walter Murray, and Margaret H. Wright. "Numerical Linear Algebra and Optimization: Volume 1." Journal of the Operational Research Society 43, no. 1 (January 1992): 74. http://dx.doi.org/10.2307/2583704.

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39

Higham, Nicholas J., and Theo Mary. "Mixed precision algorithms in numerical linear algebra." Acta Numerica 31 (May 2022): 347–414. http://dx.doi.org/10.1017/s0962492922000022.

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Today’s floating-point arithmetic landscape is broader than ever. While scientific computing has traditionally used single precision and double precision floating-point arithmetics, half precision is increasingly available in hardware and quadruple precision is supported in software. Lower precision arithmetic brings increased speed and reduced communication and energy costs, but it produces results of correspondingly low accuracy. Higher precisions are more expensive but can potentially provide great benefits, even if used sparingly. A variety of mixed precision algorithms have been developed that combine the superior performance of lower precisions with the better accuracy of higher precisions. Some of these algorithms aim to provide results of the same quality as algorithms running in a fixed precision but at a much lower cost; others use a little higher precision to improve the accuracy of an algorithm. This survey treats a broad range of mixed precision algorithms in numerical linear algebra, both direct and iterative, for problems including matrix multiplication, matrix factorization, linear systems, least squares, eigenvalue decomposition and singular value decomposition. We identify key algorithmic ideas, such as iterative refinement, adapting the precision to the data, and exploiting mixed precision block fused multiply–add operations. We also describe the possible performance benefits and explain what is known about the numerical stability of the algorithms. This survey should be useful to a wide community of researchers and practitioners who wish to develop or benefit from mixed precision numerical linear algebra algorithms.

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40

Ziegel, Eric R., and James E. Gentle. "Numerical Linear Algebra for Applications in Statistics." Technometrics 41, no. 3 (August 1999): 272. http://dx.doi.org/10.2307/1270592.

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41

Cowles, Mary Kathryn, James E. Gentle, and Kenneth Lange. "Numerical Linear Algebra for Applications in Statistics." Journal of the American Statistical Association 95, no. 450 (June 2000): 675. http://dx.doi.org/10.2307/2669416.

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42

Appa, Gautam, and Allan Findlay. "Numerical Linear Algebra and Optimization: Volume 1." Journal of the Operational Research Society 43, no. 1 (January 1992): 74–75. http://dx.doi.org/10.1057/jors.1992.12.

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43

Martinsson, Per-Gunnar, and Joel A. Tropp. "Randomized numerical linear algebra: Foundations and algorithms." Acta Numerica 29 (May 2020): 403–572. http://dx.doi.org/10.1017/s0962492920000021.

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This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues.Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.

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44

Nachaoui, Abdeljalil. "Numerical linear algebra for reconstruction inverse problems." Journal of Computational and Applied Mathematics 162, no. 1 (January 2004): 147–64. http://dx.doi.org/10.1016/j.cam.2003.08.009.

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45

G., W., and Philippe G. Ciarlet. "Introduction to Numerical Linear Algebra and Optimisation." Mathematics of Computation 55, no. 191 (July 1990): 395. http://dx.doi.org/10.2307/2008817.

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46

Sucharov, L. "Numerical linear algebra and optimization, volume 1." Advances in Engineering Software 14, no. 3 (January 1992): 237. http://dx.doi.org/10.1016/0965-9978(92)90031-a.

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47

Codenotti, B., M. Leoncini, and G. Resta. "Oracle computations in parallel numerical linear algebra." Theoretical Computer Science 127, no. 1 (May 1994): 99–121. http://dx.doi.org/10.1016/0304-3975(94)90102-3.

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48

Qin, Xiaolan, and Linzhang Lu. "Formulae for the Drazin inverse of elements in a ring." Filomat 37, no. 17 (2023): 5623–39. http://dx.doi.org/10.2298/fil2317623q.

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This paper studies additive properties of the Drazin inverse in a ring R. Some necessary and sufficient conditions for the Drazin invertible are given. Furthermore, we derive additive formulae under conditions weaker than those used in some resent papers on the subject. These extend the main results of Wei and Deng (J. Linear Multilinear Algebra, 59(12) (2011) 1319-1329) and Wang et al. (Filomat, 30(2016), 1185-1193)

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49

Taghavi, Ali, Mojtaba Nouri, Mehran Razeghi, and Vahid Darvish. "A note on non-linear ∗-Jordan derivations on ∗-algebras." Mathematica Slovaca 69, no. 3 (June 26, 2019): 639–46. http://dx.doi.org/10.1515/ms-2017-0253.

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Abstract Taghavi et al. in [TAGHAVI, A.—ROHI, H.—DARVISH, V.: Non-linear ∗-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), 426–439] proved that the map Φ: 𝓐 → 𝓐 which satisfies the following condition $$\begin{array}{} \Phi(A\diamond B)=\Phi(A)\diamond B+A\diamond \Phi(B) \end{array} $$ where A ⋄ B = AB+BA* for every A, B ∈ 𝓐 is an additive ∗-derivation. In this short note, we prove that when A is a prime ∗-algebras and Φ: 𝓐 → 𝓐 satisfies the above condition, then Φ is ∗-additive. Moreover, if Φ(iI) is self-adjoint then Φ is derivation.

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50

Brusamarello, Rosali, Érica Zancanella Fornaroli, and Ednei Aparecido Santulo. "Classification of involutions on finitary incidence algebras." International Journal of Algebra and Computation 24, no. 08 (December 2014): 1085–98. http://dx.doi.org/10.1142/s0218196714500477.

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Abstract:

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI (X), to be equivalent in the case when every multiplicative automorphism of FI (X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60 (2012) 181–188] for finitary incidence algebras.

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