Slant antieigenvalues and slant antieigenvectors of operators. (English) Zbl 1201.47037

Gustafson, Karl; Seddighin, Morteza

Linear Algebra Appl. 432, No. 5, 1348-1362 (2010).

Authors’ abstract: We introduce a general notion of slant antieigenvalue and corresponding slant antieigenvector. Then we establish how that theory may be compared to, and in some sense reduced to, the standard antieigenvalue theory. Generally speaking, our point of view is to accommodate such generalized antieigenvalue theories within the basic concepts and techniques of the original antieigenvalue theory.

Reviewer: Ali-Akbar Jafarian (West Haven)

MSC:

47B44	Linear accretive operators, dissipative operators, etc.
47A10	Spectrum, resolvent

Keywords:

slant antieigenvalues; real antieigenvalues; variational methods; convexity methods

References:

[1]	Davis, C., Extending the Kantorovich inequalities to normal matrices, Linear algebra appl., 31, 173-177, (1980) · Zbl 0434.15004
[2]	Gustafson, K., A note on the left multiplication of semigroup generators, Pacific J. math., 24, 463-465, (1968) · Zbl 0157.21405
[3]	Gustafson, K., A min – max theorem, Notices amer. math. soc., 15, 799, (1968)
[4]	Gustafson, K., Antieigenvalue inequalities in operator theory, (), 115-119
[5]	Gustafson, K., Antieigenvalues, Linear algebra appl., 208/209, 437-454, (1994) · Zbl 0815.47016
[6]	Gustafson, K., Matrix trigonometry, Linear algebra appl., 217, 117-140, (1995) · Zbl 0826.15022
[7]	Gustafson, K., Lectures on computational fluid, dynamics, mathematical physics and linear algebra, (1997), World Scientific
[8]	K. Gustafson, On geometry of statistical efficiency, 1999, Preprint.
[9]	Gustafson, K., An extended operator trigonometry, Linear algebra appl., 319, 117-135, (2000) · Zbl 0969.15013
[10]	Gustafson, K., Operator trigonometry of statistics and econometrics, Linear algebra appl., 354, 141-158, (2002) · Zbl 1015.62054
[11]	Gustafson, K., Noncommutative trigonometry, Oper. theory: adv. appl., 167, 127-155, (2006) · Zbl 1181.47007
[12]	Gustafson, K., The trigonometry of matrix statistics, Int. statist. rev., 74, 187-202, (2006)
[13]	Gustafson, K.; Rao, D., Numerical range and accretivity of operator products, J. math. anal. appl., 60, 693-702, (1977) · Zbl 0362.47001
[14]	Gustafson, K.; Rao, D., Numerical range, (1997), Springer
[15]	Gustafson, K.; Seddighin, M., Antieigenvalue bounds, J. math. anal. appl., 143, 327-340, (1989) · Zbl 0696.47004
[16]	Gustafson, K.; Seddighin, M., A note on total antiegenvectors, J. math. anal. appl., 178, 603-611, (1993) · Zbl 0803.47008
[17]	Halmos, P., A Hilbert space problem book, (1982), Springer-Verlag · Zbl 0202.12801
[18]	Hossein, Sk.M.; Paul, K.; Debnath, L.; Das, K.C., Symmetricanti-eigenvalue and symmetric anti-eigenvector, J. math. anal. appl., 345, 771-776, (2008) · Zbl 1141.47016
[19]	Khattree, R., Antieigenvalues and antieigenvectors in statistics, J. statist. plann. inference, 114, 131-144, (2003) · Zbl 1045.62064
[20]	Mirman, B., Antieigenvalues: method of estimation and calculation, Linear algebra appl., 49, 247-253, (1983) · Zbl 0529.65031
[21]	Rao, C.R., Antieigenvalues and antieigenvectors of a matrix and applications to problems in statistics, Math. inequal. appl., 10, 471-489, (2007) · Zbl 1124.15014
[22]	Seddighin, M., Antieigenvalues and total antieigenvalues of normal operators, J. math. anal. appl., 274, 239-254, (2002) · Zbl 1020.47020
[23]	Seddighin, M., Antieigenvalue techniques in statistics, Linear algebra appl., 430, 2566-2580, (2009) · Zbl 1168.15006
[24]	Seddighin, M.; Gustafson, K., On the eigenvalues which express antieigenvalues, Int. J. math. math. sci., 2005, 10, 1543-1554, (2005) · Zbl 1094.47030