CUBO A Mathematical Journal Vol.21, No02, (65–76). August 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000200065 Generalized trace pseudo-spectrum of matrix pencils Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi Department of Mathematics Faculty of Sciences of Sfax, University of Sfax Route de soukra Km 3.5, B.P. 1171, 3000, Sfax, Tunisia ammar aymen84@yahoo.fr, aref.Jeribi@fss.rnu.tn, kamelmahfoudhi72@yahoo.com ABSTRACT The objective of the study was to investigate a new notion of generalized trace pseudo- spectrum for an matrix pencils. In particular, we prove many new interesting properties of the generalized trace pseudo-spectrum. In addition, we show an analogue of the spectral mapping theorem for the generalized trace pseudo-spectrum in the matrix algebra. RESUMEN El objetivo de este estudio es investigar una nueva noción de pseudo-espectro traza ge- neralizado para pinceles de matrices. En particular, demostramos variadas propiedades nuevas e interesantes del pseudo-espectro traza generalizado. Adicionalmente, mostra- mos un análogo del teorema espectral de aplicaciones para el pseudo-espectro traza generalizado en el álgebra de matrices. Keywords and Phrases: pseudo-spectrum, condition pseudo-spectrum, trace pseudo-spectrum. 2010 AMS Mathematics Subject Classification: 15A09, 15A86, 65F40, 15A60, 65F15. http://dx.doi.org/10.4067/S0719-06462019000200065 66 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) 1 Introduction Let Mn(C) (Mn(R)) denote the algebra of all n × n complex (real) matrices, I denotes the n × n identity matrix and the conjugate transpose of U is denoted by U∗. We denote by Tr, (resp. Det) the trace (resp. determinant) map on Mn(C). In the present paper, we study the problem of finding the eigenvalues of the generalized eigenvalue problem Ux = λVx. Next, let λ ∈ C and sn(λV − U) ≤ . . . ≤ s2(λV − U) ≤ s1(λV − U) be the singular values of the matrix pencils λV −U where s1(λV −U) is the smallest and sn(λV −U) is largest singular values of the matrix pencil. Let U, V ∈ Mn(C), then the set of all eigenvalues of the matrix pencils of the form λV − U is denoted by σ(U, V) and is defined as σ(U, V) = { λ ∈ C : λV − U is not invertible } , and its spectral radius by r(U, V) = sup { |λ| : λ ∈ σ(U, V) } . For an n×n complex matrices U and V and a non-negative real number ε, the pseudo-spectrum of the matrix pencils of the form λV − U is defined as the following closed set in the complex plane σε(U, V) = { λ ∈ C : sn(λV − U) ≤ ε } . Let U, V ∈ Mn(C) and 0 < ε < 1. The condition pseudo-spectrum of the matrix pencils λV − U is denoted by Σε(U, V) and is defined as Σε(U, V) = { λ ∈ C : sn(λV − U) ≤ ε s1(λV − U) } . Let ε be a small positive number. For an operator U, V ∈ Mn(C), recall that the determinant spectrum of matrix pencils of the form λV − U is the set dε(U, V) and is defined as dε(U, V) = { λ ∈ C : |det(λV − U)| ≤ ε } . The analysis of eigenvalues and eigenvectors has had a great effect on mathematics, science, engineering, and many other fields. Then, there are countless applications for this type of analysis. The study of matrix pencils is by now a very thoughtful subject, with the notion of pseudospectrum playing a key role in the theory. However, matrix pencils play an important role in numerical linear algebra, perturbation theory, generalized eigenvalue problems. In this paper, we interest by a generalization of eigenvalues called generalized trace pseudo-spectrum for an element in the matrix CUBO 21, 2 (2019) Generalized trace pseudo-spectrum of matrix pencils 67 algebra to give more information about the matrix pencils of the form λV−U. For more information on various details on the above concepts, properties and applications of pseudo-spectrum [2, 3, 6, 7, 9], condition spectrum [1, 4, 5] and determinant spectrum [8]. Now, we introduce the new concept of the generalized trace pseudo-spectrum in the following definition. Definition 1.1. For ε > 0, the generalized trace pseudo-spectrum of the matrix pencils of the form λV − U ∈ Mn(C) is denoted by Trε(U, V) and is defined as Trε(U, V) = σ(U, V) ⋃ { λ ∈ C : |Tr(λV − U)| ≤ ε } . The generalized trace pseudoresolvent of the matrix pencils of the form λV − U is denoted by Trρε(U, V) and is defined as Trρε(U, V) = ρ(U, V) ⋂ { λ ∈ C : |Tr(λV − U)| > ε } . The singular values of a the matrix pencil are important not only for their role in diagonaliza- tion but also for their utility in a variety of applications. Since Trε(U, V) use all the singular values of λV −U to get defined, it is expected to give more information about U, V than pseudo-spectrum and condition spectrum. Since the definition use idea of ”Trace” the generalization of eigenvalues defined above is named as generalized trace pseudo-spectrum. It is easily seen that the map U → Tr(U) is continuous linear functional. Here, some important properties of the trace of U, B ∈ Mn(C) are Tr(UB) = Tr(BU), Tr(αU) = αTr(U) with α ∈ C, Tr(U + B) = Tr(U) + Tr(B). An outline of this paper is the following. In Section 2, we focuses on a new description of the generalized trace pseudo-spectra. Not only do we give a characterization of the generalized trace pseudo-spectrum in the matrix algebra. but also we investigate the connection between generalized trace pseudo-spectrum and algebraic multiplicity of the eigenvalues. In Section 3, we give an analogue of the spectral mapping theorem for the generalized trace pseudo-spectrum in the matrix algebra. 2 Generalized trace pseudo-spectrum. In this section, some relevant properties of the generalized trace pseudo-spectrum are dis- cussed in detail. For U, V ∈ Mn(C) and ε > 0, the generalized trace pseudo-spectrum of the matrix pencils of the form λV − U is denoted by Trε(U, V) and is defined as Trε(U, V) = σ(U, V) ⋃{ λ ∈ C : |Tr(λV − U)| ≤ ε } . 68 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) The generalized trace pseudo-resolvent of the matrix pencils of the form λV − U is denoted by Trρε(U, V) and is defined as Trρε(U, V) = ρ(U, V) ⋂{ λ ∈ C : |Tr(λV − U)| > ε } while the generalized trace pseudo-spectral radius of the matrix pencils of the form λV − U is defined as Trrε(U, V) := sup { |λ| : λ ∈ Trε(U, V) } . Remark 2.1. Let U, V ∈ Mn(C). Then, if V is nonsingular, then it is possible to reduce the generalized trace pseudo-spectrum to a standard trace pseudo-spectrum for the matrices V−1U or UV−1. i.e. Trε(U, V) = σ(V −1U, I) ⋃ { λ ∈ C : |Tr(λ − V−1U)| ≤ ε } , or Trε(U, V) = σ(UV −1, I) ⋃ { λ ∈ C : |Tr(λ − UV−1)| ≤ ε } . The following theorem gives some properties of the generalized trace pseudo-spectrum that follow in a straightforward manner from the definition of the generalized trace pseudo-spectrum. Theorem 2.1. Let U, V ∈ Mn(C) and ε > 0. Then, (i) Tr0(U, V) = ⋂ ε>0 Trε(U, V). (ii) If 0 < ε1 < ε2, then Trε1(U, V) ⊂ Trε2(U, V). (iii) Trε(U, V) is a non-empty compact subset of C. (iv) If α ∈ C and β ∈ C\{0}, then Trε(βU + αV, V) = βTr ε |β| (U, V) + α. (v) Trε(αV, V) = { λ ∈ C : |λ − α| ≤ ε |Tr(V)| } for all λ, α ∈ C. Proof. The proofs of items (i) and (ii) are clear from the definition of generalized trace pseudo- spectrum. (iii) Using the continuity from C to [0, ∞[ of the map λ → |Tr(λV − U)|, we get that Trε(U, V) is a compact set in the complex plane containing the eigenvalues of the matrix pencils λV − U. CUBO 21, 2 (2019) Generalized trace pseudo-spectrum of matrix pencils 69 (iv) In fact, it is well know Trε(βU + αV, V) = { λ ∈ C : |Tr(λV − βU − αV)| ≤ ε } = { λ ∈ C : |β| ∣ ∣ ∣ ∣ Tr ( λ − α β V − U ) ∣ ∣ ∣ ∣ ≤ ε } = { λ ∈ C : ∣ ∣ ∣ ∣ Tr ( λ − α β V − U ) ∣ ∣ ∣ ∣ ≤ ε |β| } . Then, λ ∈ Trε(βU + αV, V). Thus, λ − α β ∈ Tr ε |β| (U, V). Hence, λ ∈ βTr ε |β| (U, V) + α. (v) Let λ ∈ Trε(αV, V), then |Tr(λV − αV)| = |λ − α||Tr(V)| ≤ ε. This means that Trε(αV, V) = { λ ∈ C : |λ − α| ≤ ε |Tr(V)| } for all λ, α ∈ C. Q.E.D. Theorem 2.2. Let U, V ∈ Mn(C) and ε > 0. Then, (i) If U = ZBZ−1 and ZV = VZ for all nonsingular matrix Z ∈ Mn(C) we have, Trε(U, V) = Trε(B, V). (ii) If U = ZBZ−1 and V = ZKZ−1 for all nonsingular matrix Z ∈ Mn(C) we have, Trε(U, V) = Trε(B, K). (iii) The map T → Trε(U, V) is an upper semi continuous function from Mn(C) to compact subsets of C. ♦ Proof. (i) Let λ ∈ Trε(B, V), then |Tr(λV − B)| = |Tr(λV − Z−1UZ)|, = |Tr(λZ−1ZV − Z−1UZ)| = |Tr(Z−1(λZV − UZ)| = |Tr(Z−1(λV − U)Z| = |Tr(λV − U)| ≤ ε. It follows that, λ ∈ Trε(U, V). The proofs of items (ii) and (iii) follows immediately from Definition 1.1. Q.E.D. The following example shows that the converse of the assertion (i) is not true. 70 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) Example 2.1. Let U = ( 1 2 0 1 ) , B = ( 1 0 0 1 ) and V = ( 0 0 0 1 ) . Then, U and B are not similar and for ε > 0, we have Trε(U, V) = Trε(B, V) = { λ ∈ C : |λ − 2| ≤ ε } . In the following, we obtain additional results on Trε(U, V) that are useful in our analysis. Theorem 2.3. Let U, V ∈ Mn(C), λ ∈ C, and ε > 0. Then, there is D ∈ Mn(C) such that |Tr(D)| ≤ ε and Tr(λV − U − D) = 0 if, and only if, λ ∈ Trε(U, V). ♦ Proof. To see this, we suppose that there exists D ∈ Mn(C) such that |Tr(D)| ≤ ε and Tr(λV − U − D) = 0. Then, |Tr(λV − U)| = |Tr(D)| ≤ ε. Thus, λ ∈ Trε(U, V). Conversely, let λ ∈ Trε(U, V). Then, we will discuss these two cases: 1st case : If λ ∈ Tr0(U, V), then it is sufficient to take (D = 0n×n). 2nd case : λ ∈ Trε(U, V)\Tr0(U, V). Then, |Tr(λV − U)| ≤ ε. Now, we consider D = Tr(λV − U) n I. It is easy to verify that, D ∈ Mn(C) and |Tr(D)| = ∣ ∣ ∣ ∣ Tr ( Tr(λV − U) n I ) ∣ ∣ ∣ ∣ = |Tr(λV − U)| n Tr(I) ≤ ε. Also, we have Tr(λV − U − D) = Tr ( λV − U − Tr(λV − U) n I ) = 0. Q.E.D. Theorem 2.4. Let U, V ∈ Mn(C) and ε > 0. Then, Trδ(U, V) + Θε ⊆ Trε+δ(U, V), (1) holds for δ, ε > 0 with Θε, denoting the closed disk in the complex plane centered at the origin with radius ε |Tr(V)| . If we take δ = 0, we obtain an inner bound for Trε(U, V), namely Tr0(U, V) + Θε ⊆ Trε(U, V). (2) CUBO 21, 2 (2019) Generalized trace pseudo-spectrum of matrix pencils 71 Proof. Let λ ∈ Trδ(U, V) + Θε. Then, there exists there exists λ1 ∈ Trδ(U, V) and λ2 ∈ Θε such that λ = λ1 + λ2. Therefore, |Tr(λ1V − U)| ≤ δ and |Tr(λV − U)| = |Tr((λ1 + λ2)V − U)| = |Tr(λ2V) + Tr(λ1V − U)| ≤ |λ2||Tr(V)| + |Tr(λ1V − U)| ≤ |Tr(V)||λ2| + |Tr(λ1V − U)| ≤ ε + δ, so that (1) holds. Finally, let δ = 0, then the desired inclusion (2) is obtained. Q.E.D. Theorem 2.5. Let U, V ∈ Mn(C) such that UB = BU and ε > 0. If U is normal, then Trε(U + B, V) ⊆ σ(U, V) + Trε(B, V). Proof. We assume that U is normal, so there exists a unitary matrix Z ∈ Mn(C) such that Z∗UZ = λ1In1 ⊕ λ2In2 ⊕ . . . ⊕ λkInk. The condition UB = BU implies that Z∗BZ = U1 ⊕ U2 . . . ⊕ Uk where, Ui ∈ Mnk(C), i = 1, . . . , k. Then, Trε(U + B, V) = Trε(Z ∗UZ + Z∗BZ, V) = Trε((λ1In1 + U1) ⊕ . . . ⊕ (λkInk + Uk), V) = k ⋃ i=1 Trε(λiIni + Ui, V) = k ⋃ i=1 λi + Trε(Ui, V) ⊆ σ(U, V) + Trε(B, V). The proof is thus complete. Q.E.D. Remark 2.2. Let U, B and V ∈ Mn(C) and ε > 0. Then, using Theorem 2.5, we obtain the following inequality, Trrε(U + B, V) ⊆ r(U, V) + Trrε(B, V). ♦ 72 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) Theorem 2.6. Let U, B and V ∈ Mn(C) and ε > 0. Then, (i) Trε(UB, V) = Trε(BU, V). (ii) Tr ε 2 (U, V) + Tr ε 2 (B, V) ⊆ Trε(U + B, V). Proof. (i) Let λ ∈ Trε(UB, V), then ε ≥ |Tr(λV − UB)| = |Tr(λV) + Tr(−UB)| = |Tr(λV) + Tr(−BU)| = |Tr(λV − BU)|. Hence, λ ∈ Trε(BU, V). Thus, Trε(UB, V) ⊆ Trε(BU, V). The conclusion can be obtained similarly to the first inclusion, then we deduce that Trε(BU, V) = Trε(UB, V). (ii) Let λ ∈ Tr ε 2 (U, V) + Tr ε 2 (B, V). Then, there exists λ1 ∈ Tr ε 2 (U, V) and λ1 ∈ Tr ε 2 (B, V) such that λ = λ1 + λ2. Therefore, Tr(λ1V − U) ≤ ε 2 and Tr(λ2V − B) ≤ ε 2 . On the other hand, |Tr(λV − U − B)| = |Tr(λ1V − U + λ2V − B)| ≤ |Tr(λ1V − U)| + |Tr(λ2V − B)| ≤ ε Then, λ ∈ Trε(U + B, V). Q.E.D. Theorem 2.7. Let U, V ∈ Mn(C) and N ∈ Mn(C) is a nilpotent matrix and ε > 0. Then, Trε(U + N , V) = Trε(U, V). ♦ Proof. " ⊆ " Let λ ∈ Trε(U + N , V), then |Tr(λV − U − N)| ≤ ε. Since |Tr(λV − U) − Tr(N)| ≤ ε. CUBO 21, 2 (2019) Generalized trace pseudo-spectrum of matrix pencils 73 Using the fact that the matrix trace vanishes on nilpotent matrices, therefore λ ∈ Trε(U, V). Hence, Trε(U + N , V) ⊆ Trε(U, V). " ⊇ " Let λ ∈ Trε(U, V), then |Tr(λV − U)| ≤ ε. Now, we can write for any λ ∈ C |Tr(λV − U)| = |Tr(λV − U − N + N)| = |Tr(λV − U − N) + Tr(N)|. Because, Tr(N) = 0, it follows that |Tr(λV − U − N)| ≤ ε. Consequently, Trε(U, V) ⊆ Trε(U + N , V). Q.E.D. 3 Trace pseudospectral mapping Theorem Let U, V ∈ Mn(C) and f be an analytic function defined on D, an open set containing Tr0(U, V). For each ε > 0, we define ϕ(ε) = sup λ∈Trε(U,V) |Tr ( f(λ)V − f(U) ) | and suppose there exists ε0 > 0 such that Trε0(f(U), V) ⊆ f(D). Then, for 0 < ε < ε0 we define φ(ε) = sup µ ∈ f−1(Trε(U, V)) ∩ D |Tr(µV − U)|. Lemma 3.1. Let U, V ∈ Mn(C) and ε > 0, then ϕ(ε) and φ(ε) are well defined, lim ε→0 ϕ(ε) = 0 and lim ε→0 φ(ε) = 0. Proof. In the order to prove that ϕ(ε) is well defined, we define h : C → R+ h(λ) = |Tr ( f(λ)V − f(U) ) | Since h(λ) is continuous and Trε(U, V) is a compact subset of C, then it is clear that ϕ(ε) = sup { h(λ) : λ ∈ Trε(U, V) } . We conclude, ϕ(ε) is well defined. Now, let assume that there exists ε0 > 0 such that Trε0(f(U), V) ⊆ f(D). 74 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) We show that for 0 < ε < ε0, φ(ε) is well defined. Define g : C → R+, g(µ) = |Tr(µV − U)|. Since g is continuous for all µ ∈ C, then φ(ε) is well defined. It is also clear that ϕ(ε) and φ(ε) are a monotonically non-decreasing function, ϕ(ε) and φ(ε) goes to zero as ε goes to zero. Q.E.D. Theorem 3.1. Let U, V ∈ Mn(C) and let f be an analytic function defined on D, an open set containing Tr0(U, V). Then, for each f(Trε(U, V)) ⊆ Trϕ(ε)(f(U), V), where ϕ(ε) defined above. Proof. Let λ ∈ Trε(U, V). Then, using Lemma 3.1 we obtain that ϕ(ε) is well defined and lim ε→0 ϕ(ε) = 0. Therefore, h(λ) ≤ ϕ(ε). Hence |Tr ( f(λ)V − f(U) ) | := h(λ) ≤ ϕ(ε). Thus, f(λ) ∈ Trϕ(ε)(f(U), V). This means that f(Trε(U, V)) ⊆ Trϕ(ε)(f(U), V). Q.E.D. Theorem 3.2. Let U, V ∈ Mn(C) and let f be an analytic function defined on D, an open set containing Tr0(U, V). Then, for each Trε(f(U), V) ⊆ f(Trφ(ε)(U, V)). where φ(ε) defined above. Proof. Let λ ∈ Trε(f(U), V). Then, using Lemma 3.1 we obtain the existence of ε0 > 0 such that Trε(f(U), V) ⊆ Trε0(f(U), V) ⊆ f(D). Consider µ ∈ D such that λ = f(µ). Then µ ∈ f−1(Trε(U, V)), hence g(µ) ≤ φ(ε). Therefore, |Tr ( µV − U ) | := g(µ) ≤ φ(ε) Thus, µ ∈ Trφ(ε)(U, V). Then, λ = f(µ) ∈ f(Trφ(ε)(U, V)). This means that Trε(f(U), V) ⊆ f(Trφ(ε)(U, V)). Q.E.D. CUBO 21, 2 (2019) Generalized trace pseudo-spectrum of matrix pencils 75 Corollary 3.1. Combining the two inclusions in Theorems 3.1 and 3.2, we get f(Trε(U, V)) ⊆ Trϕ(ε)(f(U), V) ⊆ f(Trφ(ϕ(ε))(U, V) and Trε(f(U), V) ⊆ f(Trφ(ε)(U, V)) ⊆ Trϕ(φ(ε))(f(U), V). Here are some remarks. Remark 3.1. (i) It will be clear from the proofs of Theorems 3.1 and 3.2 that the the functions ϕ and φ measure the sizes of the trace pseudo-spectra are optimal. (ii) From the definitions of ϕ and φ, the set inclusions are sharp in the sense that the functions cannot be replaced by smaller functions. (iii) In general, the spectral mapping theorem is not true for generalized trace pseudo-spectrum. Example 3.1. Let α, β ∈ C with α 6= β 6= 0 and let U = ( α 1 0 β ) , V = ( 2 0 0 0 ) and f(λ) = λ2. Then f(U) = ( α2 α + β 0 β2 ) . A direct computation shows that Trε(f(U), V) = { λ ∈ C : |2λ − α2| ≤ ε − β2 } , f(Trε(U, V)) = { λ2 ∈ C : |2λ − α2| ≤ ε − β2 } . We can see for all ε > 0 that Trε(f(U), V) 6= f(Trε(U, V)). 76 Aymen Ammar, Aref Jeribi and Kamel Mahfoudhi CUBO 21, 2 (2019) References [1] A. Ammar, A. Jeribi and K. Mahfoudhi, A characterization of the essential approximation pseudospectrum on a Banach space, Filomath 31, (11), 3599-3610 (2017). [2] A. Ammar, A. Jeribi and K. Mahfoudhi, A characterization of the condition pseudospectrum on Banach space, Funct. Anal. Approx. Comput. 10 (2) (2018), 13–21. [3] Ammar, A., Jeribi, A., Mahfoudhi, K., The essential condition pseudospectrum and related results, J. Pseudo-Differ. Oper. Appl., (2018) 1-14. [4] Ammar, A., Jeribi, A., Mahfoudhi, K., The essential approximate pseudospectrum and related results, Filomat, 32, 6, (2018), 2139-2151. [5] Ammar, A., Jeribi, A.,Mahfoudhi, K., A characterization of Browder’s essential approxima- tion and his essential defect pseudospectrum on a Banach space, Extracta Math. ,34(1) (2019), 29-40. [6] A. Jeribi. Spectral theory and applications of linear operators and block operator matrices, Springer-Verlag, New-York, (2015). [7] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, (1991). [8] Krishna Kumar. G, Determinant spectrum: A generalization of eigenvalues, Funct. Anal. Approx. Comput. 10 (2) (2018), 1–12. [9] L. N. Trefethen and M. Embree, Spectra and pseudospectra: The behavior of nonnormal ma- trices and operators. Prin. Univ. Press, Princeton and Oxford, (2005). Introduction Generalized trace pseudo-spectrum. Trace pseudospectral mapping Theorem