On properties of real selfadjoint operators
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00101-x ORIGINAL PAPER
On properties of real selfadjoint operators Moslem Karimzadeh1 · Mehdi Radjabalipour2,3 Received: 3 February 2020 / Accepted: 6 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum 𝜎𝔽 (T) and the numerical range W𝔽 (T) of a selfadjoint operator T acting on a Hilbert space H over the real/complex field 𝔽 , and study their topological and geometrical properties which are well known in the complex case 𝔽 = ℂ . If 𝔽 = ℝ , the results are new; if 𝔽 = ℂ , the results constitute an expository body of results containing simple and short proofs of the known facts. The results are then applied to real selfadjoint operators and then to complex normal operators to sharpen their Borel functional calculi with new and shorter proofs avoiding the classical sophisticated Gelfand–Naimark theorem or the Berberian’s amalgamation theory. For such a real selfadjoint or complex normal operator N, a normed functional algebra L𝔽∞ (N) consisting of certain Borel functions defined on 𝜎𝔽 (N) is constructed which inherits the isometric properties of the continuous functional calculus f ↦ f (N) ∶ C𝔽 (𝜎𝔽 (N)) → B(H). Keywords Selfadjoint operator · Rhodius spectrum · Spectral mapping theorem · Borel functional calculus · Spectral measure Mathematics Subject Classification 47B15 · 47A10 · 47A12
Communicated by Luis Castro. * Mehdi Radjabalipour [email protected]; [email protected] Moslem Karimzadeh [email protected] 1
Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran
2
Iranian Academy of Sciences, Tehran, Iran
3
Department of Mathematics, S.B. University of Kerman, Kerman, Iran
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M. Karimzadeh and M. Radjabalipour
1 Introduction The present paper adopts the definition of Rhodius [12] for the spectrum 𝜎𝔽 (T) and continues to define the numerical range W𝔽 (T) for any bounded linear operator T acting on a Hilbert space H over the real or complex field 𝔽 ; it then proves that, in case T = T ∗ , 𝜎ℝ (T) ∩ {−||T||, ||T||} ≠ � and co(𝜎ℝ (T)) = Wℝ (T) ⊂ [−||T||, ||T||] . The results are well-known in case 𝔽 = ℂ and the independent proofs of the general case given here provide novel short proofs for the known results. This obliges us to acknowledge that the paper is partly expository. The classical properties of complex selfadjoint operators are proven here with arguments not depending on the theory of complex functions such as Liouville’s theorem in the proof of 𝜎(T) ≠ � . However, the results are not valid for real normal operators N as it is clear from the 2 × 2 real normal operator N ∶ ℝ2 → ℝ2 defined by
N[1, 0]T = [0, 1]T and N[0, 1]T = [−1, 0]T
(1.1)
that 𝜎ℝ (N) = � ≠ Wℝ (N) = {0}. In Sect. 2, certain topological as well as geometrical pr
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