Annals of Functional Analysis

Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Qingying Bu, Chingjou Liao, and Ngai-Ching Wong

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Abstract

Let $H:M_m\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\times m$ matrices. Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\{\lambda_n\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\sum_{n\geq 1} \lambda_n S^{-1}x^nS, \quad\forall x \in M_m,%\eqno{(\ddag)}$$ or $$ H(x) =\sum_{n\geq 1} \lambda_n S^{-1}(x^t)^nS, \quad\forall x \in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. In the latter case, we always have the first representation form when $H$ also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.

Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 80-89.

Dates
First available in Project Euclid: 7 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1396833505

Digital Object Identifier
doi:10.15352/afa/1396833505

Mathematical Reviews number (MathSciNet)
MR3192012

Zentralblatt MATH identifier
1308.46055

Subjects
Primary: 46G25: (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60]
Secondary: 17C65: Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] 46L05: General theory of $C^*$-algebras 47B33: Composition operators

Keywords
Holomorphic functions homogeneous polynomial orthogonally additive and multiplicative zero product preserving matrix algebras

Citation

Bu, Qingying; Liao, Chingjou; Wong, Ngai-Ching. Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices. Ann. Funct. Anal. 5 (2014), no. 2, 80--89. doi:10.15352/afa/1396833505. https://projecteuclid.org/euclid.afa/1396833505


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References

  • J. Araujo, Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity, Adv. Math. 187 (2004), 488–520.
  • W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), 199–215.
  • Y. Benyamini, S. Lassalle and J.G. Llavona, Homogeneous orthogonally additive polynomials on Banach lattices, Bull. London Math. Soc. 38 (2006), 459–469.
  • B.V. Rajarama Bhat, Linear maps respecting unitary conjugation, Banach J. Math. Anal. 5 (2011), no. 2, 1–5.
  • Q. Bu, M.-H. Hsu and N.-C. Wong, Orthogonally additive holomorphic maps between $C^*$-algebras, preprint.
  • M. Burgos, F. Fernández-Polo, J. Francisco, J.J. Garcés and A.M. Peralta, Orthogonality preservers revisited, Asian-Eur. J. Math. 2 (2009), no. 3, 387–405.
  • D. Carando, S. Lassalle and I. Zalduendo, Orthogonally additive polynomials over $C(K)$ are measures: a short proof, Integ. Eqns Operat. Theory 56 (2006), 597–602.
  • D. Carando, S. Lassalle, and I. Zalduendo; Orthogonally additive holomorphic functions of bounded type over $C(K)$, Proc. Edinburgh Math. Soc. 53 (2010), 609–618.
  • M.A. Chebotar, W.-F. Ke, P.-H. Lee and N.-C. Wong, Mappings preserving zero products, Studia Math. 155 (2003), no. 1, 77–94.
  • J.B. Conway, A course in functional analysis, second edition, Springer-Verlag New York, Inc. 1990.
  • S. Dineen, Complex analysis on infinite dimensional spaces, Springer, 1999.
  • R.E. Hartwig and P. Šemrl, Power additivity and orthogonality, SIAM J. Matrix Anal. Appl. 20 (1999), 1–13.
  • K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), 139–144.
  • J.-S. Jeang and N.-C. Wong, Weighted composition operators of $C_0(X)$'s, J. Math. Anal. Appl. 201 (1996), 981–993.
  • W.-F. Ke, B. Li and N.-C. Wong, Zero product preserving maps of continuous operator valued functions, Proc. Amer. Math. Soc. 132 (2004), 1979–1985.
  • D.H. Leung and Y.-S. Wang, Compact and weakly compact disjointness preserving operators on spaces of differentiable functions, Trans. Amer. Math. Soc. 365 (2013), 1251–1276.
  • J. Mujica, Complex analysis in Banach spaces, Math. Studies, Vol. 120, North-Holland, Amsterdam, 1986.
  • C. Palazuelos, A.M. Peralta and I. Villanueva, Orthogonally additive polynomials on $C^*$-algebras, Q.J. Math., 59 (2008), no. 3, 363–374.
  • A.M. Peralta and D. Puglisi, Orthogonally additive holomorphic functions on $C^*$-algebras, Oper. Matrices 6 (2012), no. 3, 621–629.
  • D. Perez-Garcia and I. Villanueva, Orthogonally additive polynomials on spaces of continuous functions, J. Math. Anal. Appl. 306 (2005), 97–105.