Let ℋ be a complex Hilbert space with the inner product 〈,〉 and B(ℋ) be the set of bounded linear operators on ℋ. Let ℕ be the set of all natural numbers. For the study of Jordan operators, J.W. Helton ([9] and [10]) introduced an operator T ∈ B(ℋ) which satisfies

In particular, if T is normal, then α_m(T) = (T^* − T)^m. An operator T ∈ B(ℋ) is said to be an m-symmetric operator if α_m(T) = 0. Hence T is 1-symmetric if and only if T is Hermitian. It is well known that if T is m-symmetric, then T is n-symmetric for all n ≥ m. The concept of m-symmetric operators is little strong. For example, if T is m-symmetric, then σ(T) ⊂ ℝ (cf.[10]). And T is Hermitian even if T is 2-symmetric. Also if T is normal and m-symmetric, then T is Hermitian due to the fact that T^* − T is normal and nilpotent, that is, T^* − T = 0.

Recently, C. Gu and M. Stankus ([8]) showed interesting properties of m-symmetric operators. On the other hand, for m ∈ ℕ, an operator T ∈ B(ℋ) is said to be an m-isometric operator if

It is well known that if T is m-isometric, then T is n-isometric for all n ≥ m. In 1995, J. Agler and M. Stankus [1] introduced an m-isometric operator and showed many important results of such an operator. If T is an invertible m-isometric operator and m is even, then T is (m–1)-isometric. But if T is m-symmetric and m is even, then T is always (m–1)-symmetric by Theorem 3.4 of [12]. For every odd number m, there exists an invertible m-isometric operator T which is not (m–1)-isometric (see Theorem 1 in [5]).

Throughout this paper, let I be the identity operator on ℋ and m, n be natural numbers. An operator Q ∈ B(ℋ) is said to be a nilpotent operator of order n if Qⁿ = 0 and Q^n–1 ≠ 0. For a subset A ⊂ ℂ, let A^* = {z̄ : z ∈ A }. Let σ(T) and σ_p(T) be the spectrum and the point spectrum of T ∈ B(ℋ), respectively. The approximate point spectrum of T is defined by σ_a(T) := { z ∈ ℂ : T − zI is not bounded below}, and the surjective spectrum of T is defined by σ_s(T) := { z ∈ ℂ : T − zI is not surjective}. It is known that σ(T) = σ_a(T)∪ σ_s(T), σ_a(T)^* = σ_s(T^*), and σ_s(T)^* = σ_a(T^*).

First we show the following result of m-symmetric operators.

Proposition 2.1

Let T ∈ B(ℋ). Then the following statements hold;

• T is a 2-symmetric operator if and only if T is Hermitian.

• Let T be an m-symmetric operator. For a ≠ b and non-zero vectors x, y ∈ ℋ, if Tx = ax and Ty = by, then 〈x, y〉 = 0.

• Let T be an m-symmetric operator. For a ≠ b and sequences {x_k}, {y_k} of unit vectors of ℋ, if (T − a)x_k → 0 and (T − b)y_k → 0, thenlimk→∞〈xk,yk〉=0.

Definition 1

For an operator T ∈ B(ℋ), we define γ_m,n(T) by

Then T is said to be (m, n)-isosymmetric if γ_m,n(T) = 0.

It is easy to see that

Hence if T is (m, n)-isosymmetric, then T is (m′, n′)-isosymmetric for all n′ ≥ n and n′ ≥ n. M. Stankus proved the following properties.

Proposition 2.2.([13, Corollary 30])

Let T be (m, n)-isosymmetric.

• If σ(T) ⊂ {x ∈ ℝ : |x| > 1} or σ(T) ⊂ {x ∈ ℝ : |x| < 1}, then T is n-symmetric.

• If σ(T) ⊂ {e^iθ : 0 < θ < π} or σ(T) ⊂ {e^iθ : π < θ < 2π}, then T is m-isometric.

For a, b ∈ ℂ and non-zero vectors x, y ∈ ℋ, if Tx = ax, Ty = by, then it holds that

Hence we have the following theorem.

Theorem 2.3

Let T be (m, n)-isosymmetric and x, y be unit vectors and x_k, y_k be sequences of unit vectors of ℋ.

• If Tx = ax, Ty = by, a ≠ b and a ≠ b̄, then 〈x, y〈 = 0.

• If (T − a)x_k → 0, (T − b)y_k → 0 (k → ∞), a ≠ b and a ≠ b̄, thenlimk→∞〈xk,yk〉=0.

Theorem 2.4

Let T be (m, n)-isosymmetric.

• Then T^k is (m, n)-isosymmetric for any k ∈ ℕ.

• If T is invertible, then T⁻¹is (m, n)-isosymmetric.

Theorem 2.5

Let T be (m, n)-isosymmetric and let Q be a nilpotent operator of order k. If T and Q are doubly commuting, then T +Q is (m+2k −2, n+2k −1)- isosymmetric.

Theorem 2.6

Let T be (m, n)-isosymmetric and let S be m′-isometric and n′-symmetric. If T and S are doubly commuting, then TS is (m+m′ −1, n+n′ −1)- isosymmetric.

Theorem 2.7

Let T be (m, n)-isosymmetric and let S be m′-isometric and n′-symmetric. Then T ⊗ S is (m + m′ − 1, n + n′ − 1)-isosymmetric.

In this section, we introduce [m,C]-symmetric operators and provide several examples. An antilinear operator C on ℋ is said to be a conjugation if C satisfies C² = I and 〈Cx,Cy〉= 〈y, x〉 for all x, y ∈ ℋ. An operator T ∈ B(ℋ) is said to be complex symmetric if CTC = T^* for some conjugation C.

Definition 2

For an operator T ∈ B(ℋ) and a conjugation C, we define the operator α_m(T;C) by

An operator T ∈ B(ℋ) is said to be an [m,C]-symmetric operator if α_m(T;C) = 0.

Hence if T is complex symmetric and [m,C]-symmetric, then T is m-symmetric. It holds that

Moreover, if T is [m,C]-symmetric, then T is [n,C]-symmetric for every natural number n (≥ m) and ker(α_m−1(T;C)) (m ≥ 2) is an invariant subspace for T.

Example 3.1

Let ℋ = ℂ² and let C be a conjugation on ℋ given by C(xy)=(y¯x¯) for x, y ∈ ℂ.

• If T=(i11-i) on C², then T is not Hermitian and CTC=(i11-i)=T.

  Hence T is [1, C]-symmetric.

  Hence, in this case, σ(T) = {0} due to the fact that T is nilpotent.

• Let S=(i22-i) on C². Then S is not Hermitian and CSC=(i22-i) and CSC = S ≠ S^*. Therefore, S is [1, C]-symmetric. Furthermore, σ(S) = {1,−1}.

• If R=(112i12i2) on C², then

  CR2C-2CRC·R+R2=(154-32i-32i34)-2(940094)+(3432i32i154)=0.

  Hence R is [2, C]-symmetric. It is easy to see that R is not [1, C]-symmetric. Moreover, σ(R)={32}.

• Let W=(2i11-2i) on C². Then it is easy to see that CWC = W ≠ W^*.

  Hence W is [1, C]-symmetric and σ(W)={3i,-3i}.

Example 3.2

Let ℋ = ℓ², let {en}n=1∞ be the natural basis of ℋ and let C : ℋ → ℋ be a conjugation given by

where {x_n} is a sequence in C with ∑n=1∞|xn|2<∞ and Ce_n = e_n.

• If U ∈ B(ℋ) is the unilateral shift on ℓ², then it is easy to compute U = CUC and so U is a [1, C]-symmetric operator with (unit disk).

• Let W be the weighted shift given by We_n = α_ne_n+1, where αn={2i         (n=1)n+1ni   (n≥2).

  Then

  (CW2C-2CWCW+W2)en=[(αn¯-αn)αn+1¯-αn(αn+1¯-αn+1)]en

  for all n ≥ 1. Hence W is [2, C]-symmetric operator.

An operator A ∈ ℒ(ℋ) is n-Jordan if A = T + N where T is self-adjoint, N is nilpotent of order [n+12], and TN = NT where [k] denotes the integer part of k.

Example 3.3

Let C be a conjugation C on ℋ. Suppose that A = T + N is an n-Jordan operator where T = T^* = CTC, N is nilpotent of order [n+12], TN = NT, and CN = NC. Then A is [n,C]-symmetric for the conjugation C. Indeed, since T = T^* = CTC, TN = NT, and CN = NC, it follows that

which means that A is an n-symmetric operator from [12, Theorem 3.2]. Hence A is [n,C]-symmetric.

Let C be a conjugation on ℋ. Then C satisfies ||Cx|| = ||x|| and C(αx) = ᾱ·Cx for all x ∈ ℋ and all α ∈ ℂ. Moreover, since C² = I, it follows that (CTC)^* = CT^*C and (CTC)ⁿ = CTⁿC for every positive integer n (see [7] for more details).

We now provide properties of [m,C]-symmetric operators.

Theorem 4.1

Let T ∈ B(ℋ) and let C be a conjugation on ℋ. Then the following assertions hold;

• T is an [m,C]-symmetric operator if and only if so is T^*.

• If T is an [m,C]-symmetric operator, then T^k is [m,C]-symmetric for any k ∈ ℕ.

• If T is an [m,C]-symmetric operator and invertible, then T⁻¹is [m,C]- symmetric.

• If T is a [2, C]-symmetric operator, then ker(T) ⊂ ker(T²)∩C(ker(T²)).

Lemma 4.2

For T ∈ B(ℋ), a conjugation C, and two complex numbers λ, μ, it holds

In particular, for λ ∈ ℂ we have

Proposition 4.3.([11, Lemma 3.21])

If C is a conjugation on ℋ and T ∈ B(ℋ), then σ(CTC) = σ(T)^*, σ_p(CTC) = σ_p(T)^*, σ_a(CTC) = σ_a(T)^* and σ_s(CTC) = σ_s(T)^*.

Theorem 4.4

Let T ∈ B(ℋ) be an [m,C]-symmetric operator where C is a conjugation on ℋ. Then σ_p(T) = σ_p(T)^*, σ_a(T) = σ_a(T)^*, σ_s(T) = σ_s(T)^* and σ(T) = σ(T)^*.

Lemma 4.5

Let (T,S) be a C-doubly commuting pair where C is a conjugation on ℋ. Then it holds

Theorem 4.6

Let T ∈ B(ℋ) be an [m,C]-symmetric operator and let S ∈ B(ℋ) be an [n,C]-symmetric operator where C is a conjugation on ℋ. If (T,S) is a C-doubly commuting pair, then T + S is an [m + n − 1, C]-symmetric operator.

Theorem 4.7

Let C be a conjugation on ℋ. If Q is a nilpotent operator of order n, then Q is a [2n − 1, C]-symmetric operator.

Corollary 4.8

Let T ∈ B(ℋ) be an [m,C]-symmetric operator and let Q ∈ B(ℋ) be a nilpotent operator of order n where C is a conjugation on ℋ. If (T,Q) is a C-doubly commuting pair, then T + Q is an [m+2n − 2, C]-symmetric operator.

Example 4.9

Let C_n be the conjugation on Cⁿ defined by

Assume that R_n is an n × n matrix as follows;

for a, b ∈ C. Since Q_n is nilpotent of order n, it follows that Corollary 4.8 that R_n is a [2n − 1, C_n]-symmetric operator.

Lemma 4.10

If (T,S) is a C-doubly commuting pair where C is a conjugation on ℋ, then it holds

Theorem 4.11

Let T be an [m,C]-symmetric operator and let S be an [n,C]- symmetric operator where C is a conjugation on ℋ. If (T,S) is a C-doubly commuting pair, then TS is an [m + n − 1, C]-symmetric operator.

Corollary 4.12

Let C be a conjugtion on ℋ. If T = T₁ ⊕I and S = I ⊕S₁where T₁and S₁are [m,C]-symmetric, then TS is [2m − 1, C]-symmetric.

Proposition 4.13

Let T and S be an m-isometric operator and an n-isometric operator, respectively. Then T ⊗ S is an (m + n − 1)-isometric operator.

Similarly, we show the following result.

Theorem 4.14

Let T be an [m,C]-symmetric operator and let S be an [n,D]- symmetric operator where C and D are conjugations on ℋ. Then T ⊗ S is an [m + n − 1, C ⊗ D]-symmetric operator on ℋ ⊗ ℋ.

This paper is dedicated to the memory of Professor Takayuki Furuta with cordial appreciation. This is partially supported by Grant-in-Aid Scientific Research No.15K04910. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2016R1A2B4007035).