Abstract
In this paper, we have obtained some conditions under which the boundary of the numerical range of tridiagonal matrices has no point spectrum. As a consequence of these results, the boundary of the numerical range of these tridiagonal matrices may be non-round only at the points where it touches the essential spectrum.
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References
Bilgiç H, Furkan H (2008) On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence spaces \(l_p\) and \(bv_p, (1< p<\infty )\). Nonlinear Anal 68(3):499–506
Furkan H, Bilgiç H, Başar F (2010) On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(l_p\) and \(bv_p, (1< p<\infty )\). Comput Math Appl 60(7):2141–2152
Karakaya V, Altun M (2010) Fine spectra of upper triangular double-band matrices. J Comput Appl Math 234(5):1387–1394
Chandra Tripathy B, Das R (2015) Spectrum and fine spectrum of the upper triangular matrix \(U(r, s)\) over the sequence space \(cs\). Proyecciones 34(2):107–125
Srivastava P, Kumar S (2012) Fine spectrum of the generalized difference operator \(\triangle _{v}\) on sequence space \(l_1\). Thai J Math 8(2):221–233
Karaisa A (2012) Fine spectra of upper triangular double-band matrices over the sequence space \(l_p, (1<p<\infty )\). Discrete Dyn Nat Soc 2012:1–19
Karaisa A, Başar F (2013) Fine spectra of upper triangular triple-band matrices over the sequence space \(l_p, (0<p<\infty )\). Abstr Appl Anal 2013:1–10
Dutta S, Baliarsingh P (2012) On the fine spectra of the generalized rth difference operator \(\triangle _{v}^r\) on the sequence space \(l_1\). Appl Math Comput 219(4):1776–1784
Baliarsingh P, Dutta S (2014) On a spectral classification of the operator \(\triangle _{v}^r\)r over the sequence space \(c_0\). Proc Nat Acad Sci India Sect A 84(4):555–561
Karakaya V, Manafov MD, Şimşek N (2012) On the fine spectrum of the second order difference operator over the sequence spaces \(l_p\) and \(bv_p, (1< p<\infty )\). Math Comput Model 55(3):426–431
Chien MT (1996) On the numerical range of tridiagonal operators. Linear Algebra Appl 246:203–214
Chien MT, Nakazato H (2011) The numerical range of a tridiagonal operator. J Math Anal Appl 373(1):297–304
Hansmann M (2015) On non-round points of the boundary of the numerical range and an application to non-selfadjoint Schrödinger operators. J Spectr Theory 5(4):731–750
Spitkovsky IM (2000) On the non-round points of the boundary of the numerical range. Linear Multilinear Algebra 47(1):29–33
Sims B (1974) On a connection between the numerical range and spectrum of an operator on a hilbert space. J Lond Math Soc 8:57–59
Hansmann M (2014) Absence of eigenvalues of non-selfadjoint Schrödinger operators on the boundary of their numerical range. Proc Am Math Soc 142(4):1321–1335
Hildebrandt S (1966) Über den numerischen Wertebereich eines Operators. Math Ann 163:230–247 (German)
Böttcher A, Grudsky SM (2005) Spectral properties of banded Toeplitz matrices. SIAM, Philadelphia
Kato T (2013) Perturbation theory for linear operators. Springer, Berlin
Ghoberg I, Goldberg S, Kaashoek M (1990) Classes of linear operators, vol 49. Birkhäuser, Basel
Dombrowski J (1980) Spectral properties of real parts of weighted shift operators. Indiana Univ Math J 29:249–259
Janas J, Moszyński M (2002) Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights. Integral Equ Oper Theory 43(4):397–416
Janas J, Malejki M, Mykytyuk Y (2003) Similarity and the point spectrum of some non-selfadjoint Jacobi matrices. Proc Edinb Math Soc (2) 46(03):575–595
Dombrowski J, Pedersen S (2000) Spectral properties of Jacobi matrices whose diagonal sequences are multiples of the off diagonal sequence. Math Sci Res Hotline 4(10):1–21
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Srivastava, P.D., Birbonshi, R. & Patra, A. Criteria for the Absence of Point Spectrum on the Boundary of the Numerical Range of Tridiagonal Matrices. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 245–250 (2020). https://doi.org/10.1007/s40010-018-0566-7
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DOI: https://doi.org/10.1007/s40010-018-0566-7