Opuscula Math. 38, no. 5 (2018), 623-649
https://doi.org/10.7494/OpMath.2018.38.5.623

 
Opuscula Mathematica

Spectrum of J-frame operators

Juan Giribet
Matthias Langer
Leslie Leben
Alejandra Maestripieri
Francisco Martínez Pería
Carsten Trunk

Abstract. A \(J\)-frame is a frame \(\mathcal{F}\) for a Krein space \((\mathcal{H},[\cdot,\cdot ])\) which is compatible with the indefinite inner product \([\cdot,\cdot ]\) in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in \(\mathcal{H}\). With every \(J\)-frame the so-called \(J\)-frame operator is associated, which is a self-adjoint operator in the Krein space \(\mathcal{H}\). The \(J\)-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of \(J\)-frame operators in a Krein space by a \(2\times 2\) block operator representation. The \(J\)-frame bounds of \(\mathcal{F}\) are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the \(2\times 2\) block representation. Moreover, this \(2\times 2\) block representation is utilized to obtain enclosures for the spectrum of \(J\)-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all \(J\)-frames associated with a given \(J\)-frame operator.

Keywords: frame, Krein space, block operator matrix, spectrum.

Mathematics Subject Classification: 47B50, 47A10, 46C20, 42C15.

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  • Juan Giribet
  • Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850, (1063) Buenos Aires, Argentina
  • Instituto Argentino de Matemática "Alberto P. Calderón" (CONICET), Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
  • Matthias Langer
  • University of Strathclyde, Department of Mathematics and Statistics, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
  • Leslie Leben
  • Technische Universität Ilmenau, Institut für Mathematik, Postfach 100565, D-98684 Ilmenau, Germany
  • Alejandra Maestripieri
  • Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850, (1063) Buenos Aires, Argentina
  • Instituto Argentino de Matemática "Alberto P. Calderón" (CONICET), Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
  • Francisco Martínez Pería
  • Centro de Matemática de La Plata (CeMaLP), Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172, (1900) La Plata, Argentina
  • Instituto Argentino de Matemática "Alberto P. Calderón" (CONICET), Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
  • Carsten Trunk
  • Instituto Argentino de Matemática "Alberto P. Calderón" (CONICET), Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
  • Technische Universität Ilmenau, Institut für Mathematik, Postfach 100565, D-98684 Ilmenau, Germany
  • Communicated by P.A. Cojuhari.
  • Received: 2018-02-08.
  • Accepted: 2018-03-03.
  • Published online: 2018-06-12.
Opuscula Mathematica - cover

Cite this article as:
Juan Giribet, Matthias Langer, Leslie Leben, Alejandra Maestripieri, Francisco Martínez Pería, Carsten Trunk, Spectrum of J-frame operators, Opuscula Math. 38, no. 5 (2018), 623-649, https://doi.org/10.7494/OpMath.2018.38.5.623

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