/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 499-501 ISSN 1307-5543 – www.ejpam.com Simpler Proof of the Ringrose’s Characterization of Compact Operators Aydin Sh. Shukurov Institute of Mathematics and Mechanics, NAS of Azerbaijan, Az1141, B.Vahabzade 9, Baku, Azer- baijan Abstract. The aim of this note is to give short, simpler and elementary proof of one characterization of compact operators via orthonormal sequences, which, hopefully will make this fact more accessible to nonspecialists and to a wide audience (especially to students). Beside this, the proof given here shows that this fact holds true for operator acting from Hilbert space to some Banach (not necessarily Hilbert) space. 2010 Mathematics Subject Classifications: 46B25, 47B07. Key Words and Phrases: orthonormal sequence and orthonormal bases in Hilbert space, Ringrose’s characterization of compact operators, linear operators, compact operators 1. Introduction The aim of this note is to give short, simpler and elementary proof of the following Theorem 1. Linear (not necessarily bounded) operator acting on a Hilbert space H is compact if and only if it satisfies ‖Aen‖ → 0 for each orthonormal sequence � en in H. The proof of this theorem for bounded linear operators can be found in [3]. It is shown in [2] that continuity assumption on A is superfluous. Note that, the following, in some sense more general fact is also valid: Linear (not nec- essarily bounded) operator acting on a Hilbert space H is compact if and only if it satisfies (Aen, en)→ 0 for each orthonormal sequence � en in H. For bounded operator this proposition can be found in [1, 4, 5]. It is shown in [2] that this fact remains valid without boundedness condition. Note that this proposition can be proved (which is seen from the cited references) by reducing it to the above theorem. Note that the proof of the theorem given below shows that the proposition of this theorem is also true for an operator, acting from a Hilbert space to some Banach space. Hopefully the exposition given in this short note will make the theorem and its generaliza- tion (mentioned above) more accessible to a wide audience (especially to students). Email address: ashshukurov@gmail.com http://www.ejpam.com 499 c© 2015 EJPAM All rights reserved. A. Shukurov / Eur. J. Pure Appl. Math, 8 (2015), 499-501 500 2. Main result and its proof Theorem 2. Linear (not necessarily bounded) operator acting from a Hilbert space H to some Banach space is compact if and only if it satisfies ‖Aen‖ → 0 for each orthonormal sequence � en in H. Proof. One has to prove only sufficiency of the above condition. The reverse implication is obvious (it is a well-known fact from almost all university textbooks on functional analysis, which states that compact operator takes weakly convergent sequence to convergent one). Let B be a unit ball of the space H: B = {x : ‖x‖ ≤ 1}. To prove the theorem it suffices to show that A(B) is compact. Assume the contrary: A(B) is not compact. By Hausdorff criterion there exists such an ε0 > 0 that there is not any compact ε0-net for A(B). Then there exists x1 ∈ B such that ‖Ax1‖ ≥ ε0 (otherwise 0 is ε0-net for A(B)). Without loss of generality we can take ‖x1‖ = 1. If the elements x1, x2, . . . , xn are already chosen, then (n+1)th element is determined as follows: every element x ∈ B can be represented in the form x = α1 x1 +α2 x2 + . . .+αn xn +ψ(x), where |αk| ≤ 1, for all k = 1,2, . . . , n and ψ(x) is perpendicular to x1, x2, . . . , xn. The set An = � α1Ax1 +α2Ax2 + . . .+αnAxn : |αk| ≤ 1, k = 1,2, . . . , n is compact(since it is bounded subset of finite dimensional subspace). Therefore there exists ξn+1 ∈ B such that ‖Aψ(ξn+1)‖ ≥ ε0; otherwise the relation ‖Ax − (α1Ax1 +α2Ax2 + . . .+αnAxn)‖= ‖Aψ(x)‖< ε0 would show that the compact set An is ε0-net of A(B) that contradicts the definition of ε0. De- fine xn+1 to be ψ(ξn+1) ‖ψ(ξn+1)‖ : xn+1 = ψ(ξn+1) ‖ψ(ξn+1)‖ . So, there exists orthonormal sequence x1, x2, . . . , xn such that ‖Axn‖ ≥ ε0, for all n ∈ N. Contradiction. The theorem is proved. Remark 1. Since every orthonormal sequence of a separable Hilbert space can be made an or- thonormal basis by adding new elements, it is easy to see that the following equivalent formula- tions of the above theorems holds: Proposition 1. Linear (not necessarily bounded) operator acting from a separable Hilbert space H to a Banach space B is compact if and only if it satisfies ‖Aen‖ → 0 for each orthonormal basis � en in H. ACKNOWLEDGEMENTS The author is grateful to Prof. S.S. Mirzoev for drawing his atten- tion to this question. He thanks also to I. Gahramanov for his help in finding first reference, and A.A.Huseynli for discussions. REFERENCES 501 References [1] J. H. Anderson and J. G. Stampfli. Commutators and compressions, Israel Journal of Math- ematics, 10, 433-441. 1971. [2] D. Bakic and B. Guljas. Which operators approximately annihilate orthonormal bases?, Acta Scientiarum Mathematicarum (Szeged) 64, 601-607. 1998. [3] P. A. Fillmore and J. P. Williams. On operator ranges, Advances in Mathematics 7, 254-281. 1971. [4] K. Muroi and K.Tamaki. On Ringrose’s characterization of compact operators, Mathematica Japonica 19, 259-261. 1974. [5] J. R. Ringrose. Compact non-selfadjoint operators, Van Nostrand, Princeton, 1971.