Extending the notion of a weakly positive operator introduced by Wigner, we define a weakly selfadjoint operator as the product of two selfadjoint operators, one of whose factors is positive, and show some facts on the operator

Izumino Saichi

Nakamura Noboru

University of Toyama Repository

Toyama Math. J.Vol. 30(2007), 45-56Weakly selfadjoint operatorsSaichi Izumino and Noboru NakamuraDedicated to Emeritus Professor Masahiro Nakamurain honour of his eighty-eighth birthdayAbstract. Extending the notion of a weakly positive operator in-troduced by Wigner, we define a weakly selfadjoint operator as theproduct of two selfadjoint operators, one of whose factors is positive,and show some facts on the operator.1. Throughout this note we consider bounded linear operators actingon a Hilbert space H, calling simply operators. An operator A is positiveif (Ax, x) ≥ 0 for all x ∈ H, and is denoted by A ≥ 0. By Wigner [7],a weakly positive operator was introduced as the product of two positiveinvertible operators. In [4], M. Nakamura and one of the authors studiedsuch an operator in the light of the recent knowledge. In succession ofthe consideration we define a weakly selfadjoint operator as the product oftwo selfadjoint operators, one of whose factors is positive, and show someresults on the operator.2. In the similar fashion as in [7] for a weakly positive operator, we candefine a weakly selfadjoint invertible operator T as an operator such thatT is similar to a selfadjoint operator S by some positive invertible operatorX, that is,2000 Mathematics Subject Classification. 47A10, 47A12, 47A30, 47A63.Key words and phrases. Kantorovich constant, weakly positive operator, spectralradius, weakly selfadjoint operator.4546 Saichi Izumino and Noboru Nakamura(2.1) T = XSX−1.It is easy to see that this definition is equivalent to that given as before,for, if we put A = X2 and B = X−1SX−1, then T = AB with the firstfactor A being positive and the second factor B being selfadjoint.Now first we want to state a basic fact on invertibility for a weaklyselfadjoint operator:Proposition 1. Let T = AB be a weakly selfadjoint operator. Then T isinvertible if and only if both A and B are invertible. For the spectrum σ(T )of T we have(2.2) σ(T ) = σ(A12BA12 ) if A ≥ 0,and(2.2′) σ(T ) = σ(B12AB12 ) if B ≥ 0.Proof. If T is invertible, then A(BT−1) = (BT−1)∗A = I (the identityoperator on H), so that A is invertible. Hence B = A−1T is also invertible.Conversely, if both A and B are invertible, then T is clearly invertible. For(2.2), first if A is invertible, then T = A12 (A12BA12 )A−12 , so that σ(T ) =σ(A12BA12 ). Next if A is not invertible, then T is not invertible from theabove discussion, and further easily we see that A12BA12 is not invertible.Hence 0 ∈ σ(T ), σ(A 12BA 12 ). Now from the fact σ(XY )\{0} = σ(Y X)\{0}for any operators X and Y (say, [3, p. 87]), we haveσ(T ) = [σ(A12 (A12B))\{0}]∪{0} = [σ((A 12B)A 12 )\{0}]∪{0} = σ(A 12BA 12 ).Similarly we can obtain (2.2′).Corollary 2. If T is weakly selfadjoint, then(2.3) σ(T ) ⊂ R = (−∞,∞).Related to Corollary 2 the spectrum σ(T ) of T = AB is not always in Rif we only assume that both A and B are selfadjoint, without positivity ofone factor A or B. In fact, ifWeakly selfadjoint operators 47A =[1 00 −1]and B =[−2 11 1],then T =[−2 1−1 −1]and σ(T ) ={−3±√3i2}6⊂ R.We denote by W (T ) and W (T ) the numerical range and its closure ofan operator T, respectively. Then as a fact between the spectrum andnumerical range of the product of two operators, we know the followinggeneral result due to Nakamoto [5] related to (2.3): If A or B is positivethenσ(AB) ⊂W (A)W (B).3. In [4], the following result due to Wang was shown by using the factσ(T ) ⊂ R+ = [0,∞) for a weakly positive operator T : If A ≥ 0, AB ≥ 0and ‖ B ‖≤ 1, then A ≥ AB. As an extension of this fact, we have thefollowing:Theorem 3. Let A be selfadjoint, AB ≥ 0 and ‖ B ‖≤ 1. Then |A| ≥ AB.Proof. Since Tε := (|A| + εI)−1(AB) for ε > 0 is weakly selfadjoint, wehave σ(Tε) ⊂ R. Further, we have‖ Tε ‖≤‖ (|A|+ εI)−1A ‖‖ B ‖≤ 1,so that σ(Tε) ⊂ [−1, 1]. Hence(|A|+ εI)− 12AB(|A|+ εI)− 12 ≤ r(Tε)I ≤ IorAB ≤ |A|+ εI.Taking the limit as ε→ 0, we have the desired inequality.With the similar method as in [4, Added in Proof. 2], applying mono-tonicity of the square roots of positive operators, we can give a very shortalternative proof of the above theorem: Since AB ≥ 0 and ‖ B ‖≤ 1, wesee(AB)2 = ABB∗A ≤‖ B ‖2 A2 ≤ A2,48 Saichi Izumino and Noboru Nakamuraso that by taking the square roots of both sides, we have the desired in-equality.4. Generalizing Carlin-Noble definition of the square root of a weaklypositive matrix [2], M. Nakamura and one of the authors [4] defined thefunction of a weakly positive invertible operator for every continuous func-tion defined on an interval containing its spectrum. With the similar waywe can extend the definition to any weakly selfadjoint invertible operatoras follows:Theorem 4. Let T = AB be a weakly selfadjoint invertible operator withselfajoint factors A and B. For any continuous function f defined on aninterval containing σ(T ), put(4.1) f(T ) ={A12 f(A12BA12 )A−12 if A ≥ 0,B−12 f(B12AB12 )B12 if B ≥ 0.Then f(T ) is well-defined, that is, the definition does not depend on theselfadjoint factors representing T as the product. Further, if both factorsA and B are positive, then the right two definitions of f(T ) in (4.1) areidentical.As a result, if f is defined on R, then we can define f(T ) for any weaklyselfadjoint operator T by (4.1).For the spectral mapping theorem with respect to a weakly selfadjointoperator, using the definition (4.1), we can show the following:Theorem 5. With the same assumption as in Theorem 4, the spectrumσ(f(T )) is well-defined, and σ(f(T )) = f(σ(T )).5. For a weakly positive operator the following theorem due to Bourinis well-known:Bourin’s theorem. (cf. [1], [4, Theorem 3].) Let T = AB be a weaklypositive operator with positive operators A and B. If B satisfies(5.1) mBI ≤ B ≤MBIfor some scalars 0 < mB < MB, thenWeakly selfadjoint operators 49(5.2) ‖ T ‖2≤ KBr(T )2,where r(T ) is the spectral radius of T and KB = K(mB,MB) =(MB +mB)24MBmB,called the Kantorovich constant of B with respect to (5.1).Since ‖ T ∗ ‖=‖ T ‖ and r(T ∗) = r(T ), we see that if(5.3) mAI ≤ A ≤MAIfor some scalars 0 < mA < MA, then we have(5.4) ‖ T ‖2≤ KAr(T )2,instead of (5.2). The inequality (5.2) is related to the Kantorovich constantKB with respect to the right factor B and (5.4) is related to the Kantorovichconstant KA with respect to the left factor A, so we shall often call theformer the right Bourin’s inequality, and the latter the left Bourin’s one.Now as a slight extension of Bourin’s theorem, we state the followingfact for a weakly selfadjoint operator:Theorem 6. Let T = AB be a weakly selfadjoint invertible operator.Suppose that A is positive (and invertible since T is invertible). DefineT˜ = A|B|. If mI ≤ |B| = (B2) 12 ≤ MI for some scalars 0 < m < M, then‖ T˜ ‖=‖ T ‖ and(5.5) ‖ T˜ ‖2≤ K|B|r(T˜ )2.Proof. First note that ‖ T ‖2=‖ AB2A ‖=‖ A|B|2A ‖=‖ T˜ ‖2 . For theinequality (5.5), it is immediate from the Bourin’s inequality applied to T˜ .6. In [4], a parametrized decomposition of a weakly positive operatorwas considered. We here want to discuss the similar decomposition of aweakly selfadjoint operator. Let T = AB be a weakly selfadjoint invertibleoperator. Consider the equation(6.1) ABX = XBAfor a selfadjoint X. Then we can put X = A or B−1 as a solution of (6.1),so that if we define50 Saichi Izumino and Noboru Nakamura(6.2) C(s, t) = sA+ tB−1with real parameters s and t, this operator is a little wider selfadjointsolution of (6.1). Choosing s and t such that C(s, t) is invertible, we define(6.3) A(s, t) = ABC(s, t) and B(s, t) = C(s, t)−1.Then we have a decomposition(6.4) T = A(s, t)B(s, t)of T with selfadjoint operators A(s, t) and B(s, t). If we consider the equa-tion(6.5) XAB = BAXinstead of (6.1), then since X = A−1 and B satisfy (6.5), defining C ′(s, t)by(6.6) C ′(s, t) = sA−1 + tB,we have another decomposition T = A′(s, t)B′(s, t) of T such that(6.7) A′(s, t) = C ′(s, t)−1 and B′(s, t) = C ′(s, t)AB.Now, say, corresponding to the decomposition (6.4) of T by the parame-trized factors A(s, t) and B(s, t) in (6.3), we have the following Bourin’s(left and right) inequalities: If we assume A(s, t) ≥ 0, then for T˜ =A(s, t)|B(s, t)|(6.8) ‖ T˜ ‖2=‖ T ‖2≤ KA(s,t)r(T˜ )2 (the left Bourin’s inequality)and(6.9) ‖ T˜ ‖2=‖ T ‖2≤ K|B(s,t)|r(T˜ )2 (the right Bourin’s inequality).7. We here consider the left and right Bourin’s inequalities ((6.8) and(6.9)) for the parametrized decompositions of a weakly selfadjoint operatorwith a concrete example. Since their left hand sides are constant, we onlyobserve their right hand sides. Now letWeakly selfadjoint operators 51A =[2 00 1]and B =[1 11 −4].Then A ≥ 0, T = AB =[2 21 −4], |B| = 1√29[7 −3−3 22],I ≤ A ≤ 2I, −3 +√292I ≤ |B| ≤ 3 +√292I,T˜ = A|B| = 1√29[14 −6−3 22], ‖ T˜ ‖=‖ T ‖=√20 andr(T˜ ) =18 +√34√29.For the Kantorovich constant, we then have KA = 9/8 and K|B| = 29/20.Consequently, for the left and right Bourin’s inequalities ((5.4) and (5.2))with respect to T˜ , both of the left hand sides are the same value, ‖ T˜ ‖2=‖ T ‖2= 20, and the right hand sides areKAr(T˜ )2 =9(18 +√34)2232= 22.03 · · ·andK|B|r(T˜ )2 =(18 +√34)220= 28.39 · · · ,respectively.Now consider the parametrized operators A(s, t) and B(s, t), say, in thecase (6.3). Then from (6.2), we seeC(s, t) = sA+ tB−1 =15[10s+ 4t tt 5s− t],A(s, t) = ABC(s, t) =[4s+ 2t 2s2s −4s+ t]andB(s, t) = C(s, t)−1 =1t2 − 2st− 10s2[−5s+ t tt −10s− 4t].52 Saichi Izumino and Noboru NakamuraTo apply Theorem 6, we have to assume that A(s, t) or B(s, t) is positive(and invertible). Say, assume that A(s, t) > 0. So we have to consider theleft and right Bourin’s inequalities (6.8) and (6.9) for our example. Asshown before, both the left hand sides of those (Bourin’s) inequalities havethe common value ‖ T ‖2= 20. However, the right hand sides of themdepend on the parameters s and t, and change the values as we shall show.First note that A(s, t) > 0 is equivalent to(7.1){4s+ 2t ≥ 0, −4s+ t ≥ 0detA(s, t) = 2(t2 − 2st− 10s2) > 0.If we denote by DA, the set of all points (s, t) satisfying (7.1) in thecoordinate plane, then we have(7.2) DA = D+ ∪D−,whereD+ = {(s, t); s > 0, t > (1 +√11)s}andD− = {(s, t); s < 0, t > (1−√11)s}.In this case we see that B(s, t) 6≥ 0 or detB(s, t) < 0, in fact, we see,from (6.4), (7.1) and detT = −10,detB(s, t) = detA(s, t)−1T =detTdetA(s, t)= − 10detA(s, t)< 0.By an elementary computation, we then have|B(s, t)| = 1(t2 − 2st− 10s2)√29t2 + 50st+ 25s2×[7t2 − 20st− 25s2 −3t(t+ 5s)−3t(t+ 5s) 2(11t2 + 35st+ 25s2)],T˜ (s, t) = A(s, t)|B(s, t)| = 1√29t2 + 50st+ 25s2[14t+ 10s −6t− 10s−3t+ 5s 22t+ 20s]andWeakly selfadjoint operators 53(7.3) r(T˜ (s, t)) =18t+ 15s+√34t2 + 40st− 25s2√29t2 + 50st+ 25s2Further we can obtain the following inequalities:(7.4) mA(s,t)−I ≤ A(s, t) ≤ mA(s,t)+I, m|B(s,t)|−I ≤ |B(s, t)| ≤ m|B(s,t)|+I,wheremA(s,t)± =3t±√t2 + 16st+ 80s22(> 0),andm|B(s,t)|± =±3(t+ 5s) +√29t2 + 50st+ 25s22(t2 − 2st− 10s2) (> 0).Hence the Kantorovich constants KA(s,t) and K|B(s,t)| are given as(7.5) KA(s,t) =9t28(t2 − 2st− 10s2) and K|B(s,t)| =29t2 + 50st+ 25s220(t2 − 2st− 10s2) .Now observe the right hand side of the left Bourin’s inequality. Write itRl(s, t). Then we have, from (7.3) and (7.5),(7.6) Rl(s, t) = KA(s,t)r(T˜ )2 =9t2(18t+ 15s+√34t2 + 40st− 25s2)28(t2 − 2st− 10s2)(29t2 + 50st+ 25s2)(=9t2(|18t+ 15s|+√34t2 + 40st− 25s2)28(t2 − 2st− 10s2)(29t2 + 50st+ 25s2) , because 18t+ 15s > 0)for (s, t) ∈ DA. Dividing by s4 both the numerator and denominator of thelast term of (7.6), and putting u = t/s and Rl(u) = Rl(s, t), we can rewrite(7.6) as follows:(7.7) Rl(u) =9u2(|18u+ 15|+√34u2 + 40u− 25)28(u2 − 2u− 10)(29u2 + 50u+ 25)for u > 1 +√11 or u < 1−√11.Now, for the behavior of Rl(u) we have:(i) limu→±∞Rl(u) =9(18 +√34)2232= KAr(T )2.(ii) limu→1±√11Rl(u) =∞.54 Saichi Izumino and Noboru Nakamura(iii) Rl(u) is increasing for u < 1−√11, and decreasing for u > 1+√11.Similarly, for the right Bourin’s inequality, its right hand side, denotedby Rr(s, t), is given as(7.8) Rr(s, t) = K|B(s,t)|r(T˜ )2 =(18t+ 15s+√34t2 + 40st− 25s2)220(t2 − 2st− 10s2)or, putting u = t/s and Rr(u) = Rr(s, t) in (7.8), we have(7.9) Rr(u) =(|18u+ 15|+√34u2 + 40u− 25)220(u2 − 2u− 10)for u > 1 +√11 or u < 1−√11.For the behavior of Rr(u) we have:(iv) limu→±∞Rr(u) =(18 +√34)220= K|B|r(T )2.(v) limu→1±√11Rr(u) =∞.(vi) Rr(u) is decreasing for u < −5 or u > 1 +√11, and increasing for−5 < u < 1−√11.(vii) The minimum of Rr(u) is 20, which is attained at u = −5, so that,say, for (s, t) = (−1, 5) the minimum is attained. The corresponding factorsof T areA(−1, 5) =[6 −2−2 9], B(−1, 5) = 15[2 11 −2],and then T˜ =1√5[6 −2−2 9].Finally, we want to state the outline for the case B(s, t) > 0. LetDB = {(s, t); s > 0, (1−√11)s < t < (1 +√11)s}.ThenB(s, t) > 0 is equivalent to (s, t) ∈ DB. Let us define Tˆ = |A(s, t)|B(s, t)(similarly as T˜ = A(s, t)|B(s, t)| before). Then since|A(s, t)| = 1√t2 + 16st+ 80s2[2(t2 + 10st+ 20s2) 6st6st −(t2 + 4st− 40s2)],Weakly selfadjoint operators 55we haveTˆ =1√t2 + 16st+ 80s2[2t+ 20s 2t−t 4t+ 40s].The corresponding left and right Bourin’s inequalities are‖ Tˆ ‖2= 20 ≤ K|A(s,t)|r(Tˆ )2=(3t+ 30s+√−t2 + 20st+ 100s2)2−8(t2 − 2st− 10s2)(=(3u+ 30 +√−u2 + 20u+ 100)2−8(u2 − 2u− 10))and‖ Tˆ ‖2= 20 ≤ KB(s,t)r(Tˆ )2=9(t+ 5s)2(3t+ 30s+√−t2 + 20st+ 100s2)2−20(t2 − 2st− 10s2)(t2 + 16st+ 80s2)(=9(u+ 5)2(3u+ 30 +√−u2 + 20u+ 100)2−20(u2 − 2u− 10)(u2 + 16u+ 80)),respectively, for 1−√11 < u = t/s < 1 +√11.References[1] J. Bourin, Compression, Dilation, and Matrix Inequalities, Mono-graph in RGMIA, 2004.[2] H. J. Carlin and G. A. Noble, Circuit properties of coupled dis-persive lines with applications to wave guide modeling, Proc. Networkand Signal Theory, edited by J. K. Skwirzynski and J. O. Scanlan,Peter Pergrinus, Inc., London, 1973, 258–269.[3] T. Furuta, Invitation to Linear Operators, From Matrices toBounded Linear Operators on a Hilbert Space, Taylor and Francis Inc.,New York, (2001).[4] S. Izumino and M. Nakamura,Wigner’s weakly positive operators,Scientiae Mathematicae Japonicae, 65 (2007), 1095–1102.56 Saichi Izumino and Noboru Nakamura[5] R. Nakamoto, On a theorem of Bouldin, Proc. Japan Acad., 47(1971), 979–981.[6] D. Wang, Lo¨wner-Heinz and Reid inequalities, Math. Japon., 49(1999), 247–250.[7] E. P. Wigner, On weakly positive operators, Canadian J. Math., 15(1965), 313–317.Saichi IzuminoUniversity of Toyama,Gofuku, Toyama 930-8555, Japanemail: s-izumino@h5.dion.ne.jpNoboru NakamuraFujikoshi-kogyo Senior Highschool,Higashi-ishigane 7-5, Toyama, 930-0964, Japanemail: nakamura@fujikoshiko.ac.jp(Received December 22, 2006)

Weakly selfadjoint operators

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Weakly selfadjoint operators

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