Let ||.|| be a norm on the algebra M_n of all n-by-n matrices over the
complex field C. An interesting problem in matrix theory is that "are there two
norms ||.||_1 and ||.||_2 on C^n such that ||A||=max{||Ax||_2: ||x||_1=1} for
all A in M_n. We will investigate this problem and its various aspects and will
discuss under which conditions ||.||_1=||.||_2.Comment: 8 page

C.-K. Li

G. R. Belitskii

M. Mirzavaziri

M. S. Moslehian

R. A. Horn

R. Bhatia

S. Hejazian

W. Rudin

English

arXiv

summary:Let $\Vert {\cdot }\Vert $  be a norm on the algebra ${\mathcal M}_n$ of all $n\times n$ matrices over  ${\mathbb{C}}$. An interesting problem in matrix theory is that “Are there two norms $\Vert {\cdot }\Vert _1$ and $\Vert {\cdot }\Vert _2$ on  ${\mathbb{C}}^n$ such that $\Vert A\Vert =\max \lbrace \Vert Ax\Vert _{2}\: \Vert x\Vert _{1}=1\rbrace $ for all $A\in {\mathcal M}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which $\Vert {\cdot }\Vert _1=\Vert {\cdot }\Vert _2$

Hejazian, S.

Mirzavaziri, M.

Moslehian, M. S.

Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalS. Hejazian; M. Mirzavaziri; Mohammad Sal MoslehianGeneralized induced normsCzechoslovak Mathematical Journal, Vol. 57 (2007), No. 1, 127–133Persistent URL: http://dml.cz/dmlcz/128160Terms of use:© Institute of Mathematics AS CR, 2007Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 57 (132) (2007), 127–133GENERALIZED INDUCED NORMSS. Hejazian, M. Mirzavaziri, M. S. Moslehian, Mashhad(Received October 24, 2004)Abstract. Let ‖·‖ be a norm on the algebra Mn of all n × n matrices over . Aninteresting problem in matrix theory is that “Are there two norms ‖·‖1 and ‖·‖2 on  nsuch that ‖A‖ = max{‖Ax‖2 : ‖x‖1 = 1} for all A ∈ Mn?” We will investigate thisproblem and its various aspects and will discuss some conditions under which ‖·‖1 = ‖·‖2.Keywords: induced norm, generalized induced norm, algebra norm, the full matrix alge-bra, unitarily invariant, generalized induced congruentMSC 2000 : 15A60, 47A30, 46B991. PreliminariesThroughout the paper Mn denotes the complex algebra of all n × n matricesA = [aij ] with entries in  together with the usual matrix operations. Denoteby {e1, e2, . . . en} the standard basis for  n , where ei has 1 as its ith entry and0 elsewhere. We denote by Eij the n × n matrix with 1 in the (i, j) entry and 0elsewhere.For 1 6 p 6 ∞ the `p-norm on  n is defined as follows:`p(x) = `p( n∑i=1xiei)=(n∑i=1|xi|p)1/p1 6 p < ∞,max{|x1|, . . . , |xn|} p = ∞.A norm ‖·‖ on  n is said to be unitarily invariant if ‖x‖ = ‖Ux‖ for all unitaries Uand all x ∈  n .By an algebra norm (or a matrix norm) we mean a norm ‖·‖ on Mn such that‖AB‖ 6 ‖A‖‖B‖ for all A, B ∈ Mn. An algebra norm ‖·‖ on Mn is called unitarilyThis research is partially supported by Mathematics Center of Excellence in FerdowsiUniversity.127invariant if ‖UAV ‖ = ‖A‖ for all unitaries U and V and all A ∈ Mn. See Chapter IVof [2] for more information.Example 1.1. The norm ‖A‖σ =n∑i,j=1|aij | is an algebra norm, but the norm‖A‖m = max{|ai,j | : 1 6 i, j 6 n} is not an algebra norm, since∥∥∥∥[1 11 1]2∥∥∥∥m>∥∥∥∥[1 11 1]∥∥∥∥2m.Remark 1.2. It is easy to show that for each norm ‖·‖ on Mn, the scaled normmax{‖AB‖/‖A‖‖B‖: A, B 6= 0}‖·‖ is an algebra norm; cf. [1].Let ‖·‖1 and ‖·‖2 be two norms on  n . Then for each A : (  n , ‖·‖1) → (  n , ‖·‖2)we can define ‖A‖ = max{‖Ax‖2 : ‖x‖1 = 1}. If ‖·‖1 = ‖·‖2, then ‖I‖ = 1, andthere are many examples of ‖·‖1 and ‖·‖2 such that ‖I‖ 6= 1. This shows thatgiven ‖·‖ on Mn, we cannot deduce in general that there is a norm ‖·‖1 on  n with‖A‖ = max{‖Ax‖1 : ‖x‖1 = 1}. Let us recall the concept of g-ind norm as follows.Definition 1.3. Let ‖·‖1 and ‖·‖2 be two norms on  n . Then the norm ‖·‖1,2onMn defined by ‖A‖1,2 = max{‖Ax‖2 : ‖x‖1 = 1} is called the generalized induced(or g-ind) norm constructed via ‖·‖1 and ‖·‖2. If ‖·‖1 = ‖·‖2, then ‖·‖1,1 is calledinduced norm.Example 1.4. ‖A‖C = max{ n∑i=1|ai,j | : 1 6 j 6 n}, ‖A‖R = max{ n∑j=1|ai,j | : 1 6i 6 n}and the spectral norm ‖A‖S = max{√λ : λ is an eigenvalue of A∗A} areinduced by `1, `∞ and `2, respectively.It is known that the algebra norm ‖A‖ = max{‖A‖C , ‖A‖R} is not induced andit is not hard to show that it is not g-ind too; cf. Corollary 3.2.6 of [1].We need the following proposition which is a special case of a finite dimensionalversion of the Hahn-Banach theorem [5] in which ∗ denotes the transpose; see Corol-lary 5.5.15 of [3].Proposition 1.5. Let ‖·‖ be a norm on  n and y ∈  n be a given vector. Thereexists a vector y0 ∈  n such that y∗0y = ‖y‖ and for all x ∈  n , |y∗0x| 6 ‖x‖.In this paper we examine the following nice problems:(i) Given a norm ‖·‖ on Mn. Is there any class A ofMn such that the restrictionof the norm ‖·‖ to A is g-ind?(ii) When is a g-ind norm unitarily invariant?(iii) If a given norm ‖·‖ is g-ind via ‖·‖1 and ‖·‖2, then is it possible to find ‖·‖1 and‖·‖2 explicitly in terms of ‖·‖?128(iv) When are two g-ind norms the same?(v) Is there any characterization of the g-ind norms which are algebra norms?2. Main resultsWe begin with some observations on generalized induced norms.Let ‖·‖1,2 be a generalized induced norm onMn obtained via ‖·‖1 and ‖·‖2. Then‖Eij‖1,2 = max{‖Eijx‖2 : ‖x‖1 = 1} = max{‖xjei‖2 : ‖(x1, . . . , xj , . . . , xn)‖1 =1} = αj‖ei‖2, where αj = max{|xj | : ‖(x1, . . . , xj , . . . , xn)‖1 = 1}. In general, forx ∈  n and 1 6 j 6 n, if Cx,j ∈ Mn is defined by the operator Cx,j(y) = yjx then‖Cx,j‖1,2 = αj‖x‖2.Also if for x ∈  n we define Cx ∈ Mn by Cx =n∑j=1Cx,j , then clearly ‖Cx‖1,2 =α‖x‖2, where α = max{∣∣∣n∑j=1yj∣∣∣ : ‖(y1, . . . , yj , . . . , yn)‖1 = 1}.Now we give a partial solution to Problem (i) and useful direction toward solvingProblem (iii):Proposition 2.1. Let ‖·‖ be an algebra norm on Mn. Then ‖·‖ is a g-ind normon {A ∈ Mn : ‖A‖ = ‖A−1‖ = 1}.	. Put ‖x‖1 = max{‖CAx‖ : ‖A‖ = 1}, λ−1 = max{∣∣∣n∑i=1xi∣∣∣ : ‖x‖1 =1}and ‖x‖2 = λ‖Cx‖. Then we have ‖Cy‖1,2 = max{‖Cyx‖2 : ‖x‖1 = 1} =max{∣∣∣n∑i=1xi∣∣∣‖y‖2 : ‖x‖1 = 1}= ‖y‖2λ−1 = ‖Cy‖.It follows that for each y ∈  n there is some x ∈  n such that ‖Cyx‖2 =‖Cy‖ ‖x‖1 = ‖Cy‖max{‖CDx‖ : ‖D‖ = 1}.Now let A be invertible and ‖A−1‖ = ‖A‖ = 1 and z = A−1Cyx. Thenλ−1‖Bz‖2 = λ−1‖BA−1Cyx‖2 = λ−1‖Dx‖2 = ‖CDx‖ 6 ‖Cy‖−1‖Cyx‖2 =‖Cy‖−1‖Az‖2.Now choose y so that ‖Cy‖ = 1. Then ‖CBz‖ 6 ‖CAz‖ for all B ∈ Mn. Thisimplies that ‖CAz‖ is an upper bound for the set {‖CBz‖ : ‖B‖ = 1} and indeed‖CAz‖ = max{‖CBz‖ : ‖B‖ = 1} = ‖z‖1. It follows that ‖A‖ = 1 = ‖CA(z/‖z‖1)‖ =max{‖CAu‖ : ‖u‖1 = 1} = max{‖Au‖2 : ‖u‖1 = 1} = ‖A‖1,2. Let us now answer Question (ii).129Proposition 2.2. An induced norm ‖·‖1,2 is unitarily invariant if and only if soare ‖·‖1 and ‖·‖2.	. Let U , V be unital operators and A be an arbitrary operator on  n .Suppose that ‖·‖1 and ‖·‖2 are unitarily invariant. Then‖UAV ‖1,2 = maxx6=0‖UAV x‖2‖x‖1= maxx6=0‖AV x‖2‖x‖1= maxy 6=0‖Ay‖2‖V −1y‖1= maxy 6=0‖Ay‖2‖y‖1= ‖A1,2‖.Conversely, if ‖·‖1,2 is unitarily invariant, then ‖Ux‖1 = max{‖AUx‖2 : ‖A‖1,2 61} = max{‖Bx‖2 : ‖U−1B‖1,2 6 1} = max{‖Bx‖2 : ‖B‖1,2 6 1} = ‖x‖1 and‖Ux‖2 = α−1‖CUx‖ = α−1‖UCx‖ = α−1‖Cx‖ = ‖x‖2. Modifying the proof of Theorem 5.6.18 of [3], we obtain a similar useful result forg-ind norms:Theorem 2.3. Let ‖·‖1, ‖·‖2, ‖·‖3 and ‖·‖4 be four given norms on  n andRi,j = max{ ‖x‖i‖x‖j: x 6= 0}, 1 6 i, j 6 4.Thenmax{‖A‖1,2‖A‖3,4: A 6= 0}= R2,4R3,1.In particular, max{‖A‖1,1/‖A‖2,2 : A 6= 0} = max{‖A‖2,2/‖A‖1,1 : A 6= 0} =R1,2R2,1.	. Let A be a matrix and x 6= 0. Then ‖Ax‖2/‖x‖1 = ‖Ax‖2/‖Ax‖4 ·‖Ax‖4/‖x‖3 · ‖x‖3/‖x‖1. Hence ‖A‖1,2 6 R2,4‖A‖3,4R3,1. Thus ‖A‖1,2/‖A‖3,4 6R2,4R3,1.There are vectors y, z in  n such that ‖y‖2 = ‖z‖2 = 1, ‖y‖2 = R2,4‖y‖4 and‖z‖3 = R3,1‖z‖1. By Proposition 1.5, there exists a vector z0 ∈  n such that|z∗0x| 6 ‖x‖3 and z∗0z = ‖z‖3.Put A0 = yz0. Then ‖A0z‖2/‖z‖1 = ‖yz∗0z‖2/‖z‖1 = ‖y‖2‖z‖3/‖z‖1 = ‖y‖2R3,1.Hence ‖A0‖1,2 > ‖y‖2/‖y‖4 · R3,1‖y‖4 = R2,4 · R3,1‖y‖4. On the other hand,‖A0x‖4/‖x‖3 = ‖yz∗0x‖4/‖x‖3 = ‖y‖4|z∗0x|/‖x‖3 6 ‖y‖4. Thus ‖A0‖3,4 6 ‖y‖4.Hence ‖A0‖1,2/‖A0‖3,4 > R2,4R3,1‖y‖4/‖y‖4 = R2,4R3,1. 130Corollary 2.4.(i) ‖·‖1,2 6 ‖·‖3,2 if and only if ‖·‖1 > ‖·‖3,(ii) ‖·‖1,2 6 ‖·‖1,4 if and only if ‖·‖2 6 ‖·‖4.	. (i) ‖·‖1,2 6 ‖·‖3,2 if and only if max{‖A‖1,2/‖A‖3,2 : A 6= 0} =R2,2R3,1 6 1 and this happens if and only if R3,1 6 1 or equivalently ‖·‖3 6 ‖·‖1.The proof of (ii) is similar. The following corollary completely answers Question (iv):Corollary 2.5. ‖·‖1,2 = ‖·‖3,4 if and only if there exists γ > 0 such that ‖·‖1 =γ‖·‖3 and ‖·‖2 = γ‖·‖4.	. If ‖A‖1,2 = ‖A‖3,4, then R4,2R1,3 = max{‖A‖3,4/‖A‖1,2 : A 6=0} = 1 = max{‖A‖1,2/‖A‖3,4 : A 6= 0} = R2,4R3,1. Hence max{‖x‖2/‖x‖4 : x 6=0} = R2,4 = 1/R3,1 = min{‖x‖1/‖x‖3 : x 6= 0} 6 max{‖x‖1/‖x‖3 : x 6= 0} =R1,3 = 1/R4,2 = min{‖x‖2/‖x‖4 : x 6= 0}. Thus there exists a number γ such that‖x‖2/‖x‖4 = γ = ‖x‖1/‖x‖3. Remark 2.6. It is known that each induced norm ‖·‖1,1 is minimal in the sensethat for any matrix norm ‖·‖, the inequality ‖·‖ 6 ‖·‖1,1 implies that ‖·‖ = ‖·‖1,1.But this is not true for g-ind norms in general. For instance, put ‖·‖α = `∞(·),‖·‖β = 2`2(·) and ‖·‖γ = `2(·). Then ‖·‖γ,β 6 ‖·‖α,β but ‖·‖γ,β 6= ‖·‖α,β.The following theorem is one of our main theorems and provides a complete solu-tion for Problem (v):Theorem 2.7. Let ‖·‖1 and ‖·‖2 be two norms on  n . Then ‖·‖1,2 is an algebranorm on Mn if and only if ‖·‖1 6 ‖·‖2.	. For each A and B in Mn we have‖ABx‖2 6 ‖A‖1,2‖Bx‖1 6 ‖A‖1,2‖Bx‖2 6 ‖A‖1,2‖B‖1,2‖x‖1.Hence ‖AB‖1,2 6 ‖A‖1,2‖B‖1,2.Conversely, let ‖·‖1,2 be an algebra norm. Then for each A, B ∈ Mn we have‖AB‖2 6 ‖A‖1,2‖B‖1,2‖x‖1. Let B be an arbitrary member of Mn. For Bx 6= 0,take M to be the linear span of {Bx} and define f : (M, ‖·‖1) →  by f(cBx) =c‖Bx‖1/‖Bx‖2. By the Hahn-Banach Theorem, there is an F : (  n , ‖·‖1) →  withF |M = f and ‖F‖ = ‖f‖ = max{|f(cBx)| : ‖cBx‖1 = 1} = max{|c|‖Bx‖1/‖Bx‖2 :|c| ‖Bx‖1 = 1} = 1/‖Bx‖2. Define A : (  n , ‖·‖1) → (  n , ‖·‖2) by Ay = F (y)Bx.Then ‖A‖1,2 = max{‖Ay‖2 : ‖y‖1 = 1} = max{|F (y)| ‖Bx‖2 : ‖y‖1 = 1} = 1, and131‖ABx‖2 = |F (Bx)| ‖Bx‖2 = |f(Bx)| ‖Bx‖2 = (‖Bx‖1/‖Bx‖2)‖Bx‖2 = ‖Bx‖1.Thus for all B,‖Bx‖1 = ‖ABx‖2 6 ‖A‖1,2‖B‖1,2‖x‖1 = ‖B‖1,2‖x‖1,or‖Bx‖1 6 ‖B‖1,2‖x‖1.Now take N to be the linear span of {x} and define g : (N, ‖·‖1) →  by g(cx) =c‖x‖1/‖x‖2. By the Hahn-Banach Theorem, there is a G : (  n , ‖·‖1) →  withG|N = g and ‖G‖ = ‖g‖ = max{|g(cx)| : ‖cx‖1} = max{|c| ‖x‖1/‖x‖2 : |c|‖x‖1 =1} = 1/‖x‖2. Define B : (  n , ‖·‖1) → (  n , ‖·‖2) by By = G(y)x. Then ‖B‖1,2 =max{‖By‖2 : ‖y‖1 = 1} = max{|G(y)| ‖x‖2 : ‖y‖1 = 1} = ‖x‖2‖G‖ = 1, and‖Bx‖1 = |G(x)| ‖x‖1 = |g(x)| ‖x‖1 = (‖x‖1/‖x‖2)‖x‖1 = ‖x‖21/‖x‖2.So‖x‖21‖x‖2= ‖Bx‖1 6 ‖B‖1,2‖x‖1 = ‖x‖1.Thus ‖·‖1 6 ‖·‖2. Proposition 2.8. Suppose that ‖·‖1,2 is a g-ind norm and λ > 0. Then the scalednorm λ‖·‖1,2 is a g-ind algebra norm if and only if λ > R1,2.	. Evidently, λ‖·‖1,2 = ‖·‖‖·‖1,λ‖·‖2 . If ‖·‖3,4 = λ‖·‖1,2 = ‖·‖‖·‖1,λ‖·‖2then Corollary 2.5 implies that there exists α > 0 such that ‖·‖3 = α‖·‖1 and‖·‖4 = αλ‖·‖2. Now Theorem 2.7 implies that λ‖·‖1,2 = ‖·‖3,4 is an algebra norm ifand only if α‖·‖1 6 αλ‖·‖2 or equivalently R1,2 6 λ. Proposition 2.9. Let ‖·‖1 and ‖·‖2 be two norms on  n and 0 6= α, β ∈  .Define ‖·‖α and ‖·‖β on  n by ‖x‖α = ‖αx‖1 and ‖x‖β = ‖βx‖2, respectively. Then‖·‖α and ‖·‖β are two norms on  n and ‖·‖α,β = |β/α| ‖·‖1,2.	. We have ‖A‖α,β = max{‖Ax‖β : ‖x‖α = 1} = max{‖βAx‖2 : ‖αx‖1 =1} = |β/α|max{‖Ay‖2 : ‖y‖1 = 1} = |β/α| ‖A‖1,2. The preceding proposition leads us to give the following definition:Definition 2.10. Let (‖·‖1, ‖·‖2) and (‖·‖3, ‖·‖4) be two pairs of norms on  n .We say that (‖·‖1, ‖·‖2) is generalized induced congruent (gi-congeruent) to (‖·‖3,‖·‖4) and we write (‖·‖1, ‖·‖2) ≡gi (‖·‖3, ‖·‖4) if ‖·‖1,2 = γ‖·‖3,4 for some 0 < γ ∈  .Clearly ≡gi is an equivalence relation. We denote by [(‖·‖1, ‖·‖2)]gi the equivalenceclass of (‖·‖1, ‖·‖2). Proposition 2.9 shows that for each 0 < α, β ∈  we have(α‖·‖1, β‖·‖2) ≡gi (‖·‖1, ‖·‖2). Indeed, we have the following result:132Theorem 2.11. For each pair (‖·‖1, ‖·‖2) of norms on  n we have[(‖·‖1, ‖·‖2)]gi = {(α‖·‖1, β‖·‖2) : 0 < α, β ∈  }.We can extend the above method to find some other norms on Mn which are notnecessarily gi-congruent to a given pair (‖·‖1, ‖·‖2):Proposition 2.12. Let (‖·‖1, ‖·‖2) be a pair of norms on  n and K, L ∈ Mnbe two invertible matrices. Define ‖·‖K and ‖·‖L on  n by ‖x‖K = ‖Kx‖1 and‖x‖L = ‖Lx‖2. Then ‖·‖K and ‖·‖L are norms on  n and ‖A‖K,L = ‖LAK−1‖1,2.	. Clear. See also Lemma 3.1 of [4]. Remark 2.13. Note that the case K = αI and L = βI gives Proposition 2.9.References[1] G.R. Belitskĭı, Yu. I. Lyubich: Matrix Norms and Their Applications. Operator Theory:Advances and Applications, 36. Birkhäuser-Verlag, Basel, 1988. Zbl 0645.15019[2] R. Bhatia: Matrix Analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, NewYork, 1997. Zbl 0863.15001[3] R.A. Horn, C.R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge,1994. Zbl 0801.15001[4] C.-K. Li, N.-K. Tsing, and F. Zhang: Norm hull of vectors and matrices. Linear AlgebraAppl. 257 (1997), 1–27. Zbl 0884.15023[5] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York, 1987. Zbl 0925.00005Authors’ address:           ,            , and   ! #" $ % !&   '(&)     , Department of Mathematics, Ferdowsi University, P.O. Box 1159,Mashhad 91775, Iran, e-mails: hejazian@math.um.ac.ir, mirzavaziri@math.um.ac.ir,moslehian@ferdowsi.um.ac.ir.133

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