On the Joint Spectrum for N-tuple of Hyponormal Operators

Zhang Dianzhou（张奠宙）　Huang Danrun（黄旦润）

Abstract: Let A=（A 1 ，...，A n ） be an n-tuple of double commuting hyponormal operators. It is proved that: 1. The joint spectrum of A has a Cartesian decomposition: ReSp（A）]=S p （ReA），Im[Sp（A）]=Sp（ImA）; 2. The joint resolvent of A satisfies the growth condition: ;3. If 0∉σ（Ai），i=1，2，...，n，then

If A 1 ，A 2 ，...A n are mutually commuting linear bounded operators on Hilbert space H，then the joint spectrum of n-tuples A=（A 1 ，...，A n ） can be defined in terms of the Koszul complex by J.L. Taylor. Several analysts have investigated the joint spectral properly of an n-tuple of hyponormal operators. In this paper，we shall give some new results about it，for example，the property of the Cartesian decomposition of joint spectrum，of the growth of joint resolvent，of the joint normaloid，etc.

1 Definitions and Preliminaries

We denote the Taylor joint spectrum of commuting n-tuple A=（A 1 ，...，A n ） by Sp（A，H）. We shall say that a point z=（z 1 ，...，z n ） of is in the joint approximate point spectrum σ π （A） if there exists a sequence such that

‖（A i -z i ） x k ‖→0　（k→∞），i=1，2，...，n.

We say that is in the joint compressive spectrum of A，if z∈σ x （A ＊ ），where . We denote the joint norm of A:

the joint spectral radius:

the joint numerical range:

W（A）=｛（（A 1 x，x），（A 2 x，x），...，（A n x，x））: x∈H，‖x‖=1｝

and the joint numerical radius: ω（A）=sup｛｜λ｜: λ∈W（A）｝.

If ω（A）=‖A‖，we say A is joint normaloid.

Muneo Chō has proved and ω（A）=‖A‖ iff =‖A‖ [6，7] .

Now，we quote some theorems which will be used in our discussion.

Theorem A （Taylor）. If A is a commuting n-tuple of operators，U is a neighbourhood of Sp （A），f 1 ，...，f m are analytic functions on U. Let f: U→C m be defined by f（z）=（f 1 （z），...，f m （z）） and let f（A）=（f 1 （A），...，f m （A））. Then we have

Sp（f（A），H）=f（Sp（A，H））.

Theorem B （Curto） [5] . Let H be a complex Hilbert space，A=（A 1 ，...，A n ） be an n-tuple of mutually commuting linear bounded operators， be a chain complex induced by A，where are the boundary operators. Let denote the conjugate operator of and construct an operator on as follows

Then A=（A 1 ，...，A n ） is regular in the sense of Taylor's if and only if has an inverse .

Theorem C （Curto [5] ，Corollary 3.14）. Let A=（A 1 ，...，A n ） be a commuting n-tuple ，ϕ: ｛1，...，n｝→｛1，*｝ be a function and . Assume that ϕ（A i ）ϕ（A j ）=ϕ（A j ）ϕ（A i ） for all i，j. Then Sp （ϕ（A））=｛ϕ（λ）: λ∈Sp（A）｝.

2 The Joint Spectrum of an n-tuple of Seminormal Operators

If A∈B（H），A * A-AA * ≥0，we say A is hyponormal. If A * A-AA * ≤0，we say A is cohyponormal. Operator A will be said to be seminormal，if A is either hyponormal or cohyponormal.

An n-tuple of operators A=（A 1 ，...，A n ），A i ∈B（H），will be said to be double commuting，if A i A j =A j A i ， ，i≠j，i，j=1，2，...，n.

Let A k =B k +iC k ，k=1，2，...，n，be the Cartesian decomposition of A k ∈B（H）. We denote

Re A=（Re A 1 ，...，Re A n ）=（B 1 ，...，B n ），

Im A=（Im A 1 ，...，Im A n ）=（C 1 ，...，C n ），

where A is double commuting，and so Re A and Im A are commuting n-tuples. Thus，we can define their joint spectrum.

Lemma 2.1 Let A=（A 1 ，...，A n ） be an n-tuple of normal operators，Then we have Cartesian decomposition of the joint spectrum:

Re[Sp（A）]=Sp（Re A），Im[Sp（A）]=Sp（Im A）.

Proof For any n-tuple of normal operators，it is well known that the joint spectral mapping theorem holds. Since the mappings （z 1 ，...，z n ）→（Re z 1 ，...，Re z n ），（z 1 ，...，z n ）→（Im z 1 ，...，Im z n ） are continued，we can prove this lemma by operator calculus.

Q.E.D.

Now，we recall the definition of symbol of an operator （cf.[1]）. Let T∈B（H），｛A（t）｜0≤t＜∞｝ be a contractive semigroup of operators with one parameter. Its generator is iA，i.e. A（t）=exp（iAt）. For t＜0，we set A（t）=A（-t） * . If exists，we shall call the symbol of T for A. We denote

The following theorem is a generalization of Xia's theorem （cf.[1] Ⅱ. Theorem 1.6）.

Theorem 2.2 Let A=（A 1 ，...，A n ） be a double commuting n-tuple of operators. A k =B k +iC k is the Cartesian decomposition of A k ，k =1，2，...，n. We have

（i） If ，j=1，2，...，n，then Re σx（A）⊃σ x （Re A）;

（ii） If ，j=1，2，...，n，then Im σ x （A）⊃σ x （Im A）.

Proof We confine the proof to （i），and that of （ii） is similar.

Let B=B 1 +B 2 +...+B n ，B（t）=exp（iBt）（t≥0），B（t）=B（-t） * （t＜0），B j （t）=exp（iB j t）（t≥0），B j （t）=B j （-t） * （t＜0）. Since A=（A 1 ，...，A n ） is double commuting，it is easy to see that ｛B i （t），B j （t）: i，j=1，2，...，n｝ is a commuting tuple for any t. Morever，B i （t） and B i （-t） commute with C j ，i≠j. By our present hypothesis，for each j，

For simplicity，we denote this limit by ，j=1，2，...，n.

Similarly，we can show that are also commuting tuples of normal operators （A is double commuting）. Put

Now，let b=（b 1 ，...，b n ）∈σ π （Re A）=σ π （B）=σ π [Re（B+iC ± ）]. We have Sp（A）=σ π （A） [8] ，if A is a commuting tuple of normal operators. Thus，by Lemma 2.1，there exists c=（c 1 ，...，c n ）∈Sp（C ± ） and a sequence ｛g m ｝，g m ∈H，‖g m ‖=1，m=1，2，... （or m=-1，-2，...） such that

By the definition of symbol of operators，and we can find a real number for each g m such that

Denote the class of operators which commute with B by[B]′. Since B=B 1 +...+B n is selfadjoint，we have （cf.[1]，Ⅱ，Lemma 1.1）. Thus

Let f m =exp（-it m B） g m . Then ‖f m ‖=1. Hence by （*），（**） and B j ， ，j=1，2，...，n，it follows that

Corollary 2.3 If A=（A 1 ，...，A n ） is a double commuting tuple of hyponormal operators，then

Re（σ π （A））=σ π （Re A），Im（σ π （A））=σ π （Im A）.

This result was obtained by Wei （cf.[9]） early.

Proof Since A j =B j +iC j are hyponormal，we have ，and C j ∈ ，j=1，2，...，n. （cf.[1]，Ⅱ，Theorem 2.6）. Thus，by Theorem 2.2，we have Re（σ π （A））⊃σ π （ReA），and Im（σ π （A））⊃σ π （ImA）. On the other hand，in general，σ jπ （T）=σ π （T），where T=X+iY is hyponormal，σ jπ （T）=｛λ=x+iy: ∃f n ∈H，‖f n ‖=1 such that

Theorem 2.4 If A=（A 1 ，...，A n ） is a double commuting tuple of hyponormal operators，then its joint spectrum has a Cartesian decomposition.

Proof It is sufficient to prove that Re[Sp（A）]=Sp[ReA]. From Corollary 2.3 we can see that

Sp（ReA）=σ π （ReA）=Re[σ π （A）]⊂Re[Sp（A）]，

where the first equality may be followed by the fact that Re（A） is a commuting tuple of normal operators. We shall prove Sp（ReA）⊃Re（Sp（A）） under an induction.

For n=1，the theorem holds （cf.[1]，Ⅱ，Theorem 3.2）.

For n≥2，assume that it holds for a double commuting （n-1） tuple of hyponormal operators. Then we shall prove that the theorem also holds for n. Let λ=（λ 1 ，...，λ n ）∈Sp（A）. It is well known that is not invertible [5] . The Berberian extension of A=（A 1 ，...，A n ） is denoted by [11] . It is easy to see that is also a double commuting n-tuple of hyponormal operators，and we have

Since is double commuting （n-1）-tuple of hyponormal operators，and ，we see that is not regular in the sense of Taylor's （by the assumption of the induction）.

Howerer，if we restrict the n-tuple of operators

in the subspace ，we shall see that ，0） is singular，and so is T. It is well known that the n-tuple T is regular if and only if is regular，where T=（T 1 ，...，T n ） is a normal commuting n-tuple [8] . Hence we have

Corollary 2.5 If A=（A 1 ，...，A n ） is a double commuting n-tuple of seminormal operators，then its joint spectrum has a Cartesian decomposition.

Proof There exists a function ϕ: ｛1，2，...，n｝→｛1，*｝ such that ϕ（A）=（ϕ（A 1 ），ϕ（A 2 ），...，ϕ（A n ）） is a double commution n-tuple of hyponormal operators，where . By §1，Theorem C and Theorem 2.4，we can come to the conclusion.

Definition 2.6 If A=（A 1 ，...，A n ） is a commuting n-tuple of operators，we call （A-λ） for λ∉Sp（A） the joint resolvent of A.

Lemma 2.7 （Muneo Chō） [7] If A=（A 1 ，...，A n ） is a double commuting n-tuple of hyponormal operators，then for any we have

Theorem 2.8 If A=（A 1 ，...，A n ） is a double commuting n-tuple of hyponormal operators，then for any z=（z 1 ，...，z n ）∉Sp（A），we have

‖（A-z） -1 ‖=[dist（z，Sp（A））] -1 .

Proof It is well known that （cf.[5]，p.135）

Since A=（A 1 ，...，A n ） is hyponormal，for any f: （1，2，...，n）→（0，1），we have

By Lemma 2.7，we have

This completes the proof.

Q.E.D.

Now，we consider the seminormal operators. Let A=（A 1 ，...，A n ） be a double commuting n-tuple of seminormal operators，ϕ: （1，2，...，n）→（1，*），and ϕ（A i ）= . We set ϕ（A）=（ϕ（A 1 ），...，ϕ（A n ））. Curto [5] showed that Sp（ϕ（A），H）=｛ϕ（λ）: λ∈Sp（A）｝. If p is a permutation of （1，2，...，n），p（A）=（A p（1） ，...，A p（n） ），then ，U，V are unitary operators （cf.[5]，p.137）. Thus， . If A exsists，then p（A） also has an inverse （cf.[3]）.

Theorem 2.9 If A=（A 1 ，...，A n ） is a double commuting n-tuple of seminormal operators，then for any z=（z 1 ，...，z n ）∉Sp（A），we have

3 The Normaloid Property of an n-Tuple of Semi-hyponormal Operators

Theorem 3.1 If A=（A 1 ，...，A n ） is a double commuting n-tuple of semi-hyponormal operators，then Sp（A）=σ ρ （A），where σ ρ （A） is the joint compressive spectrum of A.

Q.E.D.

Muneo Chō and Makoto Takaguchi have proved that every double commuting n-tuples of hyponormal operators satisfies r sp （A）=‖A‖. In the case of semi-hyponormal operators，we conjecture that it remains true. But we now only prove two particular cases.

Corollary 3.2 If A=（A 1 ，...，A n ） is a double commuting n-tuple of semi-hyponormal operators，A i =U i ｜A i ｜，where U i are unitary，dim ，i=1，2，...，n，then

r Sp （A）=‖A‖.

Proof By Theorem 3.1，it is easy to see that A-Z is invertible iff is invertible. For any ，we can find Z i ∈σ（A i ） such that （cf.[1]，Ⅱ，Theorem 3.3）. Now，we may apply the proof which was used by M. Chō and M. Takaguchi [7] to the case of semi-hyponormal operators. Then the assertion will be proved.

Q.E.D.

If B is an isometric operator，we set

iff there is ε 1 ＞0 such that

iff there is ε 2 ＞0 such that

Then we can establish the lemma.

Theorem 3.4 If T=（T 1 ，T 2 ，...，T n ） is a double commuting n-tuple of semi-hyponormal operators，and if 0∉σ（T i ） i=1，2，...，n，then we have

Proof Let Ti=U i ｜T i ｜ be the polar decomposition of T i ，i=1，2，...，n. Since 0∉σ（T i ），we see that Ui is unitary，and ｜T i ｜ is invertible. It is easy to see that （U 1 ，...，U n ） and（｜T 1 ｜，...，｜T n ｜） are double commuting and U i ｜T j ｜=｜T j ｜U i ，i≠j. Let U=U 1 ...U n ，where U is unitary. Since T i are semi-hyponormal，we have

Thus ‖T‖=‖｜T｜‖.

Now，since ｜T｜=（｜T 1 ｜，...，｜T n ｜） is a commuting n-tuple of normal operators，we have ω（｜T｜）=‖｜T｜‖=‖ T‖ [6] . Put . Then by the inequlity ，i=1，2，...，n. On the other hand

Thus

Finally，if T=（T 1 ，T 2 ，...，T n ） is a double commuting n-tuple of operators，where T i are semi-hyponormal or semi-cohyponormal，then Corollary 3.2 and Theorem 3.4 are also true. The proofs are omitted.

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