Research Article Characterization of Numerical Ranges of Posinormal

International Journal of Modern Science and Technology Vol. 2, No. 3, 2017. Page 85-89. http://www.ijmst.co/ ISSN: 2456-0235.

Research Article

Characterization of Numerical Ranges of Posinormal Operator S. Asamba1, R. K. Obogi1, N. B. Okelo2,* 1Department of Mathematics, Kisii University, P.O. Box 408-40200, Kisii. Kenya. 2School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O. Box 210-40601, Bondo-Kenya. *Corresponding author’s e-mail: [email protected] Abstract Let be a complex Hilbert space equipped with the inner product ; and let be the algebra of bounded linear operators acting on . The numerical range of a bounded linear operator on a complex Hilbert space is the set The numerical radius of is given by . In this paper we investigate the numerical range of an operator acting on a complex Hilbert space. In particular, we characterize the numerical range of a posinormal operator on an infinite dimensional complex Hilbert space. The present paper shows that for a posinormal operator A, W(A) is nonempty, always and is an ellipse whose foci are the eigenvalues of A. Keywords: Numerical range; Linear operator; Posinormal operator; Hilbert Space. Introduction operator such that ; where Let be a Hilbert space equipped with the is called the interrupter [6]. denotes the inner product ; and let be the algebra set of all posinormal operators on is said to of bounded linear operators acting on . The be coposinormal if is also posinormal. The numerical range of a bounded linear operator general problem being considered is, given a on a complex Hilbert space is the set posinormal operator what can be said [1]. about its numerical range [7]. Preliminaries The numerical radius of is given by It follows that In the present section, some basic properties is the image of the unit circle in under of numerical ranges of operators which are to be the quadratic form from to used in later discussions are given. and is the smallest radius of a circular disk Proposition 2. 1: Let Then by [8] centered at the origin which contains [2]. 1. The study of numerical ranges has along and 2. contains all of the eigenvalues of extensive history. Toeplitz was the first to introduce the notion of the numerical range [3]. 3. is contained in the closed disk of The famous Toeplitz- Hausdorff theorem radius around the origin. gives the most fundamental property of the 4. If then numerical range that the numerical range of all bounded linear operators is always convex [4]. It implies that the quadratic form maps 5. If is unitary then the unit circle of H to a subset of with all its interior filled up[5]. In this article we will 6. if and only if is self-adjoint. concentrate in characterizing the numerical range 7. of a posinormal operator. An operator is said to be posinormal if there exists a positive

Received: 21.02.2017; Received after Revision: 04.03.2017; Accepted: 05.03.2017; Published: 28.03.2017 ©International Journal of Modern Science and Technology. All rights reserved. 85 Asamba et al., 2017. Characterization of Numerical Ranges of Posinormal Operator

Theorem 2.2: For , either characterize the numerical range of posinormal i. If then operators [10]. Spectral theory of linear ii. If the eigenvalues of are equal and is operators on Hilbert spaces is a pillar in several developments in mathematics, physics and not a multiple of identity, is a non- quantum mechanics. Its concepts like the trivial closed disk centered at the spectrum of a linear operator, eigenvalues and eigenvalues of A. vectors, spectral radius, spectral integrals among Theorem 2.3: Let have trace zero. others have useful applications in quantum Then A is unitarily equivalent to a matrix whose mechanics, a reason why there is a lot of current diagonal entries are all zero. research on these concepts and their Theorem 2.4: (The Folk Theorem) Let generalizations. Spectral theory is described as a be such that If no closed disk of rich and important theory as it relates perfectly contains , then 휆 is an eigenvalue of A. with other areas including measure and Proposition 2.5: For operators A and B on integration theory and theory of analytic spaces H and K respectively, the following hold: functions. i. W(A) is nonempty bounded convex subset of ℂ. If H is finite dimensional, Results and discussions then W(A) is even compact. Lemma 4.1: Let H be a complex Hilbert space ii. for any and B(H) the algebra of all bounded linear complex a and b. operators on H. Let AB(H) be posinormal then iii. for any unitary W(A) is an ellipse whose foci are the operator on H. eigenvalues of A [11]. iv. If A is unitarily equivalent to an operator Proof. Choose A such that A= with of the form (that is, A is a and as the eigenvalues of A. Now if , we have dialation of B or B is a compression of A ), then A – 휆I= - . v. A complex number z is in W(A) if and Let x= ( ), then only if A is unitarily equivalent to an . operator of of the form Therefore, vi.

Methodology Hence the radius is Therefore the For a successful completion of this paper, numerical range W . It thus background knowledge of Functional analysis, the operator theory, especially normal operators, follows that W is a circle with the center at 휆 self adjoint operators, hyponormal operators on a and radius . [12] Hilbert space, numerical range and the spectrum Now if and a we have of operators on a Hilbert space is vital. We have stated some known fundamental principles A If then which shall be useful in our research. The methodology involved the use of known . inequalities and techniques like Cauchy - Therefore taking the inner product Ax, x we Schwartz inequality, Minkowski’s inequality, get parallelogram law and the polarization identity. Lastly, we used the technical approach of tensor Ax, x products in solving the stated problem. Also the methodology involved the use of known inequalities and techniques like the polarization identity [9]. We have also used the technical So . approach of derivatives of fuzzy sets to

Now letting , we therefore write the and therefore normalizing the vector x we see above equation as follows. that Ax, x since Now we have That is[14] So W(A) is the set of convex combinations of and and is the segment joining them. If and 0 we choose This implies that such that it lies between and . We therefore From the definition of an operator norm, have,

. Therefore, In this case, we let and Hence, W(A) is a circle of radius centered at so, (0,0). Lemma 4.3: Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. Let A B(H) be posinormal then Here we see that W(A’) is an ellipse with the W(A) is nonempty. center at (0,0) and the minor axis , and foci at Proof. Let be orthonormal sequence of (r,0) and (-r,0). Thus, W(A) is an ellipse with vectors in H. For to exist in H then foci at and and the major axis has an inclination of 휃 with the real axis. The sequence is bounded and The following example, illustrates on how to because has norm 1. Now, using calculate the numerical range of any given A=A* (because all posinormal operators are self- operator A on a Hilbert space H. adjoint and positive) we have Example 4.2: In let be the operator defined by matrix . Take , , with

and

Therefore, as n→ the sequence Taking the absolute values on both sides we converges weakly to 0 in H such that have . Thus x is an eigenvector for the eigenvalue . This implies that W (A) is nonempty.

The next result due to Toeplitz and Hausdorff So a circle of radius shows that the numerical range of all bounded centered at linear operators acting on the Hilbert space H is Alternatively, given the operator A defined by always convex. We give its proof for matrix completion. Theorem 4.4: Let H be a complex Hilbert space We then have the characteristic polynomial and A B(H) be posinormal, then W (A) is given by always convex. Proof. Let , . We prove and hence finding the characteristic equation we that whenever . see that Therefore, is the If where are such that eigenvalue. Since for the norm we have and it is sufficient to show that for all . Let us fix unit vectors such that ,

and define by Shapiro among others. These operators [15] are , of great importance since they are vital in Moreover, there exists such that formulation of principles of mathematical Since analysis and quantum mechanics. The operators Now observe that the vectors x and y are linearly include normal operators, posinormal operators, independent. Otherwise for some α , hyponormal operators, normaloid operators and among others. Posinormal operators have not been exhaustively characterized despite the fact Define continuous functions z and f by that they possess interesting properties. Certain properties of posinormal operators have been , and characterized for example continuity and . linearity but numerical ranges and spectra of A straight forward calculation shows that f is posinormal operators have not been considered. real-valued function with and Also the relationship between the numerical Implying that range and spectrum has not been determined for as posinormal operators. Therefore the objectives required. of this study were: to investigate numerical Corollary 4.5: Let A B(H) be posinormal then ranges of posinormal operator, to investigate the spectra of posinormal operators and to establish Proof. Since A is bounded, then every the relationship between the numerical range and eigenvalue of A that lies on the boundary of spectrum of a posinormal operator. The W(A) is a normal eigenvalue. An eigenvalue 휆 is methodology involved use of known inequalities said to be normal for an operator if like Cauchy- Schwartz inequality, Minkowski’s inequality, the parallelogram law and the polarization identity to determine the numerical Let us assume without loss of generality that range and spectrum of posinormal operators and λ Suppose there is a unit vector for which our technical approach shall involve use of but . Let . tensor products. The results obtained shall be Because the pair used in classification of Hilbert space operators is orthonormal in H, and therefore spans a and shall be applied in other fields like quantum two dimensional subspace M. It follows that information theory to optimize minimal output W(A) contains the numerical range of the entropy of quantum channel; to detect compression of A to M. It is enough to show entanglement using positive maps ;and for local that 0 is in the interior of . Now the distinguishability of unitary operators.The matrix of with respect to the orthonormal numerical range of a bounded posinormal operator A acting on a complex Hilbert space H basis of is of the form where is an ellipse whose foci are the eigenvalues of A. . We need to show that this For a posinormal operator A , the will establish as a non-degenerate following properties hold: The numerical range elliptical disk with one focus at 0, and therefore of A is nonempty, i.e ; is complete the proof. Now, always convex; Zero is contained in the numerical range of A, i.e The norm Where the term on the right, upon recalling that of A is a subset of the closure of W(A), , is just . The numerical radius of A is equal to the norm of A, i.e . Conflict of interest as desired. Authors declare there are no conflicts of interest. Conclusions References Hilbert space operators have been studied by [1] Okelo B, Kisengo S. On the numerical many mathematicians including Hilbert, Weyl, range and spectrum of normal operators on Neumann, Toeplitz, Hausdorff, Rhaly, Mecheri,

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