Trapezoid Orthogonality in Complex Normed Linear Spaces

Abstract

Let $G_p(x,y,z)={\parallel x+y+z\parallel }^p+{\parallel z\parallel }^p{-\parallel x+z\parallel }^p-{\parallel y+z\parallel }^p$ be defined on a normed space $\mathcal{X}$. The special case $G_2(x,y,z)=0,\forall z\in \mathcal{X}$, where $\mathcal{X}$ is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where $\mathcal{X}$ is a complex inner product space endowed with the inner product $⟨·,·⟩$ and induced norm $\parallel ·\parallel$, it is proved that $Sgn\left(G_2(x,y,z)\right)=Sgn\left(Re⟨x,y⟩\right),\forall z\in \mathcal{X}$, and a geometric explanation for condition $Re⟨x,y⟩=0$ is provided. Furthermore, a condition $G_2(x,iy,z)=0,\forall z\in \mathcal{X}$ is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.

1. Introductions and Preliminaries

Orthogonality was originally a concept closely related to inner product spaces, but it was later extended to more general normed spaces. In this article, we mainly study a class of orthogonality in complex inner product spaces and normed linear spaces. Prior to this, although some concepts are already widely known, let us first review them for more accurate expressions in the following discussion.

1.1. Preliminaries

Throughout this article, we denote by $\mathbb{R}$ and $\mathbb{C}$ the fields of real and complex numbers, respectively. The single letter i denotes the imaginary unit which satisfies $i^2=-1$. A series of basic definitions and notations are as follows.

Definition 1 

([1]). Given $\alpha =a+bi\in \mathbb{C}$, where $a,b\in \mathbb{R}$.

1. 

The complex number $a-bi$ is called the conjugate of α, denoted by $\overline{\alpha }$.

2. 

The numbers a and b are the real part and the imaginary part of α, respectively. We write them as $a=Re\left(\alpha \right)$, $b=Im\left(\alpha \right)$.

3. 

The absolute value $\left|\alpha \right|$ is the non-negative root of $\alpha \overline{\alpha }$, i.e., $\left|\alpha \right|=\sqrt{a^2+b^2}$. ($\left|\alpha \right|$ is also referred to as the modulus of α in some studies.)

If $\alpha =a+bi\in \mathbb{C}$ and $\alpha \ne 0$, and let $\lambda ={\displaystyle \frac{\overline{\alpha }}{a^2+b^2}}$, then it is easy to see that $\lambda \alpha =\alpha \lambda =1$. At this point, we refer to $\lambda$ as the inverse of $\alpha$ and represent it as ${\alpha }^{-1}$ or ${\displaystyle \frac{1}{\alpha }}$; and we often write ${\alpha }^{-1}\beta$ as ${\displaystyle \frac{\beta }{\alpha }}$. As a special case, it is evident that

$$i^{-1}\equiv {\displaystyle \frac{1}{i}}=-i.$$

(1)

The bold letter $\mathbf{0}$ will always represent a zero vector in the corresponding space.

Definition 2 

([2,3]). Case I: An inner product in a linear space $\mathcal{H}$ over a scalar field $\mathbb{F}=\mathbb{R}$ is a real valued function of each ordered pair of vectors $x,y\in \mathcal{H}$, denoted as $\langle x,y\rangle$, having the following properties:

  (i) 

Symmetry: $\langle y,x\rangle =\langle x,y\rangle$.

 (ii) 

Bilinearity: $\langle x,y\rangle$ is a linear function of x for fiexed y, and a linear function of y for fixed x.

(iii) 

Positivity: $\langle x,x\rangle \ge 0$ for all $x\in \mathcal{H}$; and $\langle x,x\rangle =0$ if and only if $x=\mathbf{0}$.

Case II: When the scalar field $\mathbb{F}=\mathbb{C}$, function $\langle x,y\rangle$ is complex valued and the above property (iii) still holds true, but properties (i) and (ii) are altered, as follows:

(i’) 

Skew symmetry: $\langle y,x\rangle =\overline{\langle x,y\rangle }$.

(ii’) 

Sesquilinearity: $\langle x,y\rangle$ is a linear function of x for fixed y, and a skewlinear function of y for fixed x, satisfying $\langle \alpha x,y\rangle =\alpha \langle x,y\rangle$ and $\langle x,\alpha y\rangle =\overline{\alpha }\langle x,y\rangle$.

A linear space $\mathcal{H}$ with an inner product is called an inner product space, abbreviated as i.p.s. An inner product space is called a real inner product space/complex inner product space, abbreviated as r.i.p.s./c.i.p.s., if the inner product is defined as Case I/Case II of Definition 2.

Obviously, Definition 1 and the skew symmetry of Definition 2 yield

$$Re\langle x,y\rangle ={\displaystyle \frac{1}{2}}\left(\langle x,y\rangle +\langle y,x\rangle \right)$$

(2)

and

$$Im\langle x,y\rangle ={\displaystyle \frac{1}{2i}}\left(\langle x,y\rangle -\langle y,x\rangle \right)\stackrel{\left(1\right)}{=}{\displaystyle \frac{-i}{2}}\left(\langle x,y\rangle -\langle y,x\rangle \right).$$

(3)

Under Definition 2, the traditional orthogonality can be defined as follows.

Definition 3 

([2]). Two vectors x and y are called orthogonal if $\langle x,y\rangle =0$, written as $x\perp y$.

Definition 4 

([2,3]). A linear space $\mathcal{X}$ over a field $\mathbb{F}\in \left\{\mathbb{R},\mathbb{C}\right\}$ is said to be a normed linear space if, to every $x\in \mathcal{X}$, a non-negative real number $\parallel x\parallel$ is associated, called the norm of x, in such a way that

• Subadditivity: $\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$, for all $x,y\in \mathcal{X}$.
• Homogeneity: $\parallel \alpha x\parallel =\left|\alpha \right|\parallel x\parallel$, for all $x\in \mathcal{X}$ and $\alpha \in \mathbb{F}$.
• Positivity: $\parallel x\parallel >0$, if $x\ne \mathbf{0}$; and $\parallel \mathbf{0}\parallel =0$.

A normed linear space is called a real normed linear space when $\mathbb{F}=\mathbb{R}$, and called a complex normed linear space when $\mathbb{F}=\mathbb{C}$. A normed linear space $\mathcal{X}$ endowed with the norm $\parallel ·\parallel$ is usually denoted by $(\mathcal{X},\parallel ·\parallel )$. Perhaps the most commonly discussed normed linear spaces are $L_p$ and $l_p$ spaces, whose specific definitions can be found in, e.g., [2] (pp. 39–40).

It is well known that every i.p.s. can be normed by defining $\parallel x\parallel =\sqrt{\langle x,x\rangle }$, which is usually called the norm induced by the inner product $\langle ·,·\rangle$. However, a general normed linear space may not necessarily be endowed with any inner product. Therefore, the so-called “orthogonality” cannot be directly established in general normed linear spaces by the concept defined in Definition 3.

1.2. An Introduction to Some Types of Orthogonality and Related Theorems

A large number of studies have been devoted to the study of various types of orthogonality, such as Roberts orthogonality [4], Birkhoff orthogonality (also known as Birkhoff–James orthogonality) [5,6,7], isosceles orthogonality (also known as James orthogonality) [6], Pythagorean orthogonality [6], $\alpha$-orthogonality [8], ${\perp }_{\rho }$ orthogonality (also known as norm derivatives orthogonality) [9,10], etc., in normed spaces, which are generally not inner product spaces. Some detailed reviews of related research can be found in the literature, such as [10,11,12].

Some types of orthogonality suitable for extending to complex normed linear spaces are listed below.

Definition 5. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a normed linear space over $\mathbb{F}$, where $\mathbb{F}\in \left\{\mathbb{R},\mathbb{C}\right\}$. For any two vectors $x,y\in \mathcal{X}$,

1. 

x is said to be Pythagorean orthogonal to y (written as $x{\perp }_{P}y$) if ${\parallel x+y\parallel }^2={\parallel x\parallel }^2+{\parallel y\parallel }^2.$

2. 

x is said to be isosceles orthogonal to y (written as $x{\perp }_{I}y$) if $\parallel x+y\parallel =\parallel x-y\parallel$ when $\mathcal{X}$ is a real normed space or $\left\{\begin{array}{c}\parallel x+y\parallel =\parallel x-y\parallel ,\hfill \\ \parallel x+iy\parallel =\parallel x-iy\parallel \hfill \end{array}\right.$ when $\mathcal{X}$ is a complex normed space (cf. [13]).

3. 

x is said to be Birkhoff orthogonal to y (written as $x{\perp }_{B}y$) if the inequality $\parallel x+\alpha y\parallel \ge \parallel x\parallel$ holds for any $\alpha \in \mathbb{F}$.

4. 

x is said to be Roberts orthogonal to y (written as $x{\perp }_{R}y$) if the equality $\parallel x+\alpha y\parallel =\parallel x-\alpha y\parallel$ holds for any $\alpha \in \mathbb{F}$.

(The trapezoid orthogonality in real normed linear spaces and complex normed linear spaces that we will focus on discussing will be stated in Definitions 6 and 9, respectively.) Among the various types of orthogonality listed in Definition 5, it is widely recognized that the Roberts orthogonality is the strongest. For example, Roberts orthogonality implies Birkhoff orthogonality (see, e.g., [9,10,13,14,15,16] or inequality (31)) but the converse generally does not hold true.

Theories related to the above concepts have been widely studied and have undergone new developments in many fields (see, e.g., [13,16,17,18,19,20,21,22,23]).

There are two important theorems that will be useful in the upcoming discussion.

Theorem 1 

(Clarkson inequalities, part of Th. 2 of [24]). Let $p\in (1,\infty ]$ and a normed space $\mathcal{X}\in \left\{L_p,l_p\right\}$. For two arbitrary elements $x,y\in \mathcal{X}$,

$$\begin{array}{cc}\hfill & {2(\parallel x\parallel }_p^p+{\parallel y\parallel }_p^p{)\le \parallel x+y\parallel }_p^p{+\parallel x-y\parallel }_p^p\le 2^{p-1}{(\parallel x\parallel }_p^p+{\parallel y\parallel }_p^p),ifp\ge 2;\hfill \\ \hfill & {2(\parallel x\parallel }_p^p+{\parallel y\parallel }_p^p{)\ge \parallel x+y\parallel }_p^p{+\parallel x-y\parallel }_p^p\ge 2^{p-1}{(\parallel x\parallel }_p^p+{\parallel y\parallel }_p^p),if1<p\le 2,\hfill \end{array}$$

where ${\parallel ·\parallel }_p$ denotes the $L_p$-norm or $l_p$-norm in the corresponding spaces.

The special case of $p=2$ for the Clarkson inequalities leads to

$${\parallel x+y\parallel }_2^2{+\parallel x-y\parallel }_2^2={2\parallel x\parallel }_2^2+2{\parallel y\parallel }_2^2,forallx,y\in \mathcal{X}\in \{L_2,l_2\}.$$

(4)

It is commendable that, just one year before the publication of the Clarkson inequalities, Jordan and von Neumann had proposed the famous “parallelogram law”, as follows.

Theorem 2 

([25], Th. I). A normed linear space $(\mathcal{X},\parallel ·\parallel )$ can be an inner product space if and only if

$${\parallel x+y\parallel }^2{+\parallel x-y\parallel }^2={2\parallel x\parallel }^2+2{\parallel y\parallel }^2,forallx,y\in \mathcal{X}.$$

Theorem 2 has become a cornerstone of later research on the theory of i.p.s., see e.g., [11,26,27].

1.3. Some Basic Properties of a Class of Three-Variable Functions

Let $(\mathcal{X},\parallel ·\parallel )$ be a normed linear space over a field $\mathbb{F}$. Let us consider the three-variable function $G_p:\mathcal{X}\times \mathcal{X}\times \mathcal{X}\to \mathbb{R}$ defined by

$$G_p(x,y,z)={\parallel x+y+z\parallel }^p+{\parallel z\parallel }^p{-\parallel x+z\parallel }^p-{\parallel y+z\parallel }^p,$$

(5)

where the power exponent $p\in [1,\infty )$. Some basic properties of $G_p(x,y,z)$ are summarized without proofs in the following proposition.

Proposition 3. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a normed linear space over $\mathbb{F}$ and function $G_p(x,y,z)$ defined as (5). Then, for any given $p\in [1,\infty )$ and all $x,y,z,x_1,x_2\in \mathcal{X}$, the following conclusions are true:

(a) 

$G_p(x,x,x)\ge 0$; and $G_p(x,x,x)=0$, for $p\in (1,\infty )$, if and only if $x=\mathbf{0}$.

(b) 

$G_p(\mathbf{0},y,z)=G_p(x,\mathbf{0},z)=0$.

(c) 

$G_p(x,y,z)=G_p(y,x,z)$.

At times, we use the symbol “∀” to express “for all”.

Some further characterizations of the equalities $G_p(x,y,z)=0,\forall z\in \mathcal{X}$ will be presented in Section 3 (see Lemma 11). For the case where the power exponent p in (5) takes some special values, we have the following basic results.

Proposition 4. 

Let $\mathcal{X}\in \left\{L_p,l_p\right\}$ and function $G_p(x,y,z)$ be defined as (5), with $\parallel ·\parallel ={\parallel ·\parallel }_p$ representing the $L_p$-norm or $l_p$-norm. Then, the following are true:

(a) 

$G_p(x,x,z)\ge 0$, for all $x,z\in \mathcal{X}$ and $p=1,2$.

(b) 

$G_p(x,-x,z)\le 0$, for all $x,z\in \mathcal{X}$ and $p=1,2$.

(c) 

$G_1(x,y,\pm (x+y))\le 0$, for all $x,y\in \mathcal{X}$.

Proof. 

Notice that $G_p(x,x,z)={\parallel 2x+z\parallel }_p^p+{\parallel z\parallel }_p^p-2{\parallel x+z\parallel }_p^p$. Thus, the inequality $G_1(x,x,z)\ge 0$ can be directly proven according to the triangle inequality. For the case that $p=2$, we have

$${\parallel 2x+z\parallel }_2^2+{\parallel z\parallel }_2^2\stackrel{\left(4\right)}{=}{\displaystyle \frac{1}{2}}{(\parallel 2x+2z\parallel }_2^2+{\parallel 2x\parallel }_2^2{)\ge 2\parallel x+z\parallel }_2^2,$$

which yields $G_2(x,x,z)\ge 0$. Therefore, property (a) is proven.

Since $G_p(x,-x,z)={2\parallel z\parallel }_p^p{-(\parallel x+z\parallel }_p^p{+\parallel x-z\parallel }_p^p)$, then inequality $G_1(x,-x,z)\le 0$ can be directly derived from the triangle inequality. For the case $p=2$, we have

$$\begin{array}{c}\hfill {\parallel x+z\parallel }_2^2{+\parallel x-z\parallel }_2^2\stackrel{\left(4\right)}{=}{2\parallel x\parallel }_2^2+{2\parallel z\parallel }_2^2\ge 2{\parallel z\parallel }_2^2,\end{array}$$

thus obtaining property (b).

Property (c) can be directly derived from the triangle inequality. □

Let $Sgn(·)$ denote the sign of a given real value. In the following discussion, we always express the i.p.s. $\mathcal{H}$ endowed with the inner product $\langle ·,·\rangle$ and induced norm $\parallel ·\parallel =\sqrt{\langle ·,·\rangle }$ as $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$. Also, let $G_2(x,y,z)$, which is the case where $p=2$ in (5), be a function defined on $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$.

For the r.i.p.s., some results involving $G_2(·,·,·)$ have been established as follows.

Theorem 5 

([28], Proposition 1). Let $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ be an r.i.p.s. and $x,y\in \mathcal{H}$. Then,

$$Sgn\left(G_2(x,y,z)\right)=Sgn\left(\langle x,y\rangle \right),foranyz\in \mathcal{H},$$

(6)

where $G_2(x,y,z)$ is defined by (5).

1.4. Trapezoid Orthogonality (T-Orthogonality) in Real Normed Linear Spaces

Theorem 5 implies that, for a given $x,y$ in an r.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$,

$$G_2(x,y,z)=0,\forall z\in \mathcal{H}\iff \langle x,y\rangle =0.$$

(7)

Inspired by (7), a question naturally arises.

Question 1. 

For a general normed space $\mathcal{X}$ (which may not be an i.p.s.), can the following equations

$$G_p(x,y,z)=0,forallz\in \mathcal{X}$$

(8)

be applied to characterize the orthogonality between x and y?

Interestingly, we belatedly learned that $p=2$ for the equalities in (8) had been studied by Alsina, Cruells, and Tomàs in [9] to define the trapezoid orthogonality (T-orthogonality) in real normed linear spaces.

Definition 6 

([9]). (To our knowledge, this concept was later generalized by Ger [29] to the φ-orthogonality.) Let $(\mathcal{X},||·\parallel )$ be a real normed linear space and $x,y\in \mathcal{X}$. We say x T-orthogonal to y (written as $x{\perp }^{T}y$) if the equations

$$G_2(x,y,z)=0,forallz\in \mathcal{X}$$

(9)

hold.

Reference [9] has shown that T-orthogonality is stronger than some types of orthogonality.

Theorem 6 

([9], Theorems 2.2, 2.3). Let $(\mathcal{X},||·\parallel )$ be a real normed linear space and $x,y\in \mathcal{X}$. Then, the following are true:

1. 

$x{\perp }^{T}y\Rightarrow x{\perp }_{P}y$; and $\mathcal{X}$ is an i.p.s. if for all $x,y\in \mathcal{X}$, $x{\perp }_{P}y\Rightarrow x{\perp }^{T}y$.

2. 

$x{\perp }^{T}y\Rightarrow x{\perp }_{I}y$; and $\mathcal{X}$ is an i.p.s. if for all $x,y\in \mathcal{X}$, $x{\perp }_{I}y\Rightarrow x{\perp }^{T}y$.

3. 

$x{\perp }^{T}y\Rightarrow x{\perp }_{B}y$; and $\mathcal{X}$ is an i.p.s. if for all $x,y\in \mathcal{X}$, $x{\perp }_{B}y\Rightarrow x{\perp }^{T}y$.

However, we have to note that the original version of T-orthogonality proposed by [9] was based on a property of isosceles trapezoid in the real plane, which was proven by [30]. So, the authors of [9] naturally only studied the results in real normed linear spaces. Additionally, it is regrettable that the comparison between T-orthogonality and Roberts orthogonality seems to have been omitted in [9].

In this article, we will extend some related research to complex normed linear spaces.

1.5. The Organization of the Remaining Parts of This Article

The rest of this article is organized as follows. In Section 2, we extend the results with the form as (6) from r.i.p.s. to c.i.p.s. (in Proposition 7) and provide a geometric explanation for the constraint conditions (in Corollary 9). In Section 3.2, we first introduce the T-orthogonality in complex normed linear spaces by adding the condition $G_2(x,iy,z)=0,\forall z\in \mathcal{X}$, which is indispensable as demonstrated in the subsequent argument. Some characterizations of T-orthogonality, such as symmetry, linear independence, additivity, homogeneity, etc., are also studied. Furthermore, in Section 3.3, we compare the T-orthogonality with some well-known types of orthogonality, and subsequently reveal (in Proposition 14) that T-orthogonality implies Roberts orthogonality. In Section 3.4, we study the existence of complex normed linear spaces with T-orthogonal vectors as i.p.s. (or not).

It should be noted that, only in terms of substantive contribution, Propositions 3, 4, and Lemma 11 may be somewhat redundant, as this article only focuses on the case of $p=2$. However, realizing that these conclusions may have some potential applications, we are still willing to report them to readers.

2. Results in Complex Inner Product Spaces

The ultimate aim of this section is to explain why adding condition (20) to express the trapezoid orthogonality in complex normed linear spaces is reasonable. As a byproduct, we first extend result (6) to the c.i.p.s. and attempt to provide a geometric explanation for condition (12).

Proposition 7. 

Given a c.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ and let $x,y\in \mathcal{H}$. Then, for any fixed $z\in \mathcal{H}$,

$$Sgn\left(G_2(x,y,z)\right)=Sgn\left(Re\langle x,y\rangle \right),$$

(10)

where $G_2(x,y,z)$ is defined by (5).

Proof. 

Since

$$\begin{array}{cc}\hfill & G_2(x,y,z)={\parallel x+z+y\parallel }^2+{\parallel z\parallel }^2{-\parallel x+z\parallel }^2-{\parallel y+z\parallel }^2\hfill \\ \hfill & ={\parallel x+z\parallel }^2+\langle x+z,y\rangle +\langle y,x+z\rangle +{\parallel y\parallel }^2+{\parallel z\parallel }^2\hfill \\ \hfill & {-\parallel x+z\parallel }^2-{\parallel y\parallel }^2-\langle y,z\rangle -\langle z,y\rangle -{\parallel z\parallel }^2\hfill \\ \hfill & =\langle x,y\rangle +\langle y,x\rangle \stackrel{\left(2\right)}{=}2Re\langle x,y\rangle ,\hfill \end{array}$$

then, we obtain (10). □

Corollary 8. 

Given a c.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ and let $x,y\in \mathcal{H}$. Then, equalities

$$G_2(x,y,z)=0,\forall z\in \mathcal{H}$$

(11)

hold true if and only if

$$Re\langle x,y\rangle =0.$$

(12)

Proposition 7 is naturally an extension of Theorem 5.

Day ([26] Th. 7.2) has pointed out that $Re\langle x,y\rangle$ is a real inner product (in the r.i.p.s.) if and only if $\langle x,y\rangle$ is a complex inner product (in the corresponding c.i.p.s.). In fact, given a c.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ and let $x,y\in \mathcal{H}$, the following properties can be derived from (2):

  (i)

For all $x\in \mathcal{H}$, $Re\langle x,x\rangle \ge 0$, and $Re\langle x,x\rangle =0$ if and only if $x=\mathbf{0}$.

 (ii)

For all $x,y\in \mathcal{H}$, $Re\langle x,y\rangle =Re\langle y,x\rangle$.

(iii)

For all $x_1,x_2,y\in \mathcal{H}$, $Re\langle x_1+x_2,y\rangle =Re\langle x_1,y\rangle +Re\langle x_2,y\rangle$.

(iv)

For all $x,y\in \mathcal{H}$ and $\alpha ,\beta \in \mathbb{R}$, $Re\langle \alpha x,\beta y\rangle =\alpha \beta Re\langle x,y\rangle$.

 (v)

For all $x,y\in \mathcal{H}$, $|Re\langle x,y\rangle |\le \parallel x\parallel \parallel y\parallel$.

Based on the above results, especially properties (iv) and (v) of $Re\langle x,y\rangle$, we attempt to introduce the following definition.

Definition 7. 

(Just before submitting this manuscript, we learned that a similar definition had been proposed by Poon [22].) Let $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ be a c.i.p.s. and $x,y\in \mathcal{H}-\left\{\mathbf{0}\right\}$. We say that

$$\widehat{[x,y]}=arccos\left({\displaystyle \frac{Re\langle x,y\rangle }{\parallel x\parallel \parallel y\parallel }}\right),\widehat{[x,y]}\in [0,\pi ],$$

(13)

is the real angle between x and y. We specify $\widehat{[x,y]}=\pi /2$ when $x=\mathbf{0}$ or $y=\mathbf{0}$.

Therefore, the identity in (10) can be restated as

$$G_2(x,y,z)\left\{\begin{array}{cc}\ge 0,\hfill & if\widehat{[x,y]}\in [0,\pi /2],\hfill \\ \le 0,\hfill & if\widehat{[x,y]}\in [\pi /2,\pi ].\hfill \end{array}\right.$$

(14)

Hence, we can translate Corollary 8 into the following statement.

Corollary 9. 

Given a c.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ and let $x,y\in \mathcal{H}$. Then, $\widehat{[x,y]}=\pi /2$ if and only if the equalities in (11) hold.

However, recall that the usual “angle” between two vectors should be defined as follows. (The following definition is well known and can be found in many basic textbooks.)

Definition 8. 

Let $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ be a c.i.p.s. and $x,y\in \mathcal{H}-\left\{\mathbf{0}\right\}$. We say that

$$\angle [x,y]=arccos\left({\displaystyle \frac{\langle x,y\rangle }{\parallel x\parallel \parallel y\parallel }}\right),\angle [x,y]\in [0,\pi ],$$

is the angle between x and y. We specify $\angle [x,y]=\pi /2$ when $x=\mathbf{0}$ or $y=\mathbf{0}$.

Inspired by the above discussion, we have come to the following conclusion.

Proposition 10. 

Given a c.i.p.s. $(\mathcal{H},\langle ·,·\rangle ,\parallel ·\parallel )$ and let $x,y\in \mathcal{H}$. Then, $\angle [x,y]=\pi /2$ if and only if equalities in (11) and the following equalities

$$G_2(x,iy,z)=0,\forall z\in \mathcal{H}$$

(15)

all hold true.

Proof. 

According to Corollary 8, equalities (11) and (15) are equivalent to

$$\left\{\begin{array}{c}Re\langle x,y\rangle =0,\\ Re\langle x,iy\rangle =0.\end{array}\right.$$

(16)

Since

$$\begin{array}{c}\hfill Im\langle x,y\rangle \stackrel{\left(3\right)}{=}{\displaystyle \frac{-i}{2}}\left(\langle x,y\rangle -\langle y,x\rangle \right)\stackrel{Sesquilinearity}{========}{\displaystyle \frac{1}{2}}\left(\langle x,iy\rangle +\langle iy,x\rangle \right)\stackrel{\left(2\right)}{=}Re\langle x,iy\rangle ,\end{array}$$

we then see that $\langle x,y\rangle =0$ if and only if equalities in (16) both hold true, and complete this proof. □

Corollary 9 and Proposition 10 characterize the difference between the orthogonality in the sense of a “real angle” and the usual angle. Furthermore, we will show that a condition similar to (15), namely (20), will play an important role in the subsequent discussion in Section 3.2.

3. Some Characterizations of Trapezoid Orthogonality in Complex Normed Linear Spaces

Next, we will study trapezoid orthogonality in general (complex) normed linear spaces, which may not necessarily have inner products.

3.1. Some Results on Equalities Involving $G_p(x,y,z)$

In order to investigate the T-orthogonality using the function $G_2(x,y,z)$, we will first present some results for the more general function $G_p(x,y,z)$, which are also the follow-up works of Proposition 3.

Let $\mathbb{N}$ denote the set of positive integers, and $\mathbb{Q}$ the field of rational numbers.

Lemma 11. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a normed linear space over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and function $G_p(x,y,z)$ defined as (5). Then, for any given $p\in [1,\infty )$ and $x,y,x_1,x_2,y_1,y_2\in \mathcal{X}$, the following are true:

(a) 

For any $\alpha \in \mathbb{C}-\left\{0\right\}$, $G_p(\alpha x,y,z)=0$, $\forall z\in \mathcal{X}$ if and only if $G_p(x,{\displaystyle \frac{1}{\alpha }}y,z)=0$, $\forall z\in \mathcal{X}$.

(b) 

If $G_p(x_k,y,z)=0$, $\forall z\in \mathcal{X}$, $k=1,2$, then $G_p(x_1+x_2,y,z)=0$, $\forall z\in \mathcal{X}$.

If $G_p(x,y_k,z)=0$, $\forall z\in \mathcal{X}$, $k=1,2$, then $G_p(x,y_1+y_2,z)=0$, $\forall z\in \mathcal{X}$.

(c) 

If $G_p(x,y,z)=0$, $\forall z\in \mathcal{X}$, then

(c.1) 

$G_p(\alpha x,\alpha y,z)=0$, $\forall z\in \mathcal{X}$, $\forall \alpha \in \mathbb{F}=\mathbb{C}$.

(c.2) 

$G_p(x,-y,z)=G_p(-x,y,z)=0$, $\forall z\in \mathcal{X}$.

(c.3) 

$G_p(\lambda x,\mu y,z)=0$, $\forall z\in \mathcal{X}$, $\forall \lambda ,\mu \in \mathbb{F}=\mathbb{R}$.

Proof. 

Properties (a) and (c.1) are clearly verifiable due to the arbitrariness of z.

To prove property (b), rewrite $G_p(x_2,y,z)=0,\forall z\in \mathcal{X}$ by replacing z with $z+x_1$, i.e.,

$$\parallel x_2+y+z+x_1{\parallel }^p+\parallel z+x_1{\parallel }^p-\parallel x_2+z+x_1{\parallel }^p-{\parallel y+z+x_1\parallel }^p=0,\forall z\in \mathcal{X}.$$

(17)

Add (17) to $G_p(x_1,y,z)=0,\forall z\in \mathcal{X}$ to obtain $G_p(x_1+x_2,y,z)=0,\forall z\in \mathcal{X}$. The proof for the other part $G_p(x,y_1+y_2,z)=0,\forall z\in \mathcal{X}$ is similar to the above process.

To prove property (c.2), replacing z with $z-y$ for $G_p(x,y,z)=0,\forall z\in \mathcal{X}$ actually yields $G_p(x,-y,z)=0,\forall z\in \mathcal{X}$, which, upon combining with property (a) (take $\alpha =-1$) yields $G_p(-x,y,z)=0,\forall z\in \mathcal{X}$.

To prove property (c.3), first, under the assumption that $G_p(x,y,z)=0,\forall z\in \mathcal{X}$, based on property (b), we iterate the variables located at x and y to see that equalities

$$G_p(\lambda x,\mu y,z)=0,\forall z\in \mathcal{X}$$

(18)

hold for all $\lambda ,\mu \in \mathbb{N}$. Furthermore, combining (18) with properties (a) and (c.2), it is easy to see that equalities in (18) still hold for all $\lambda ,\mu \in \mathbb{Q}$. Finally, from the fact that function $G_p(x,y,z)$ is continuous and $\mathbb{Q}$ is dense in $\mathbb{R}$, we obtain property (c.3). □

3.2. Trapezoid Orthogonality (T-Orthogonality) in Complex Normed Linear Spaces

Proposition 10 implies that the combination of equalities (11) and (15) coincides with the orthogonality in the c.i.p.s. Inspired by this, we introduce the trapezoid orthogonality (T-orthogonality) in complex normed linear spaces as follows.

Definition 9. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a complex normed linear space with function $G_2(·,·,·)$ defined by (5) and $x,y\in \mathcal{X}$. We say x T-orthogonal to y (written as $x{\perp }^{T}y$) if all the following equalities

$$G_2(x,y,z)=0,\forall z\in \mathcal{X},$$

(19)

$$G_2(x,iy,z)=0,\forall z\in \mathcal{X},$$

(20)

hold true. And x and y are said to be a couple of T-orthogonal vectors if $x{\perp }^{T}y$.

Some fundamental properties of T-orthogonality are summarized as follows.

Proposition 12. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a complex normed linear space. The following are true:

  (i) 

Trivial orthogonality: For all $x,y\in \mathcal{X}$, $\mathbf{0}{\perp }^{T}y$, and $x{\perp }^T\mathbf{0}$.

 (ii) 

Non-degeneracy: For all $x\in \mathcal{X}$, $x{\perp }^{T}x$ if and only if $x=\mathbf{0}$.

(iii) 

Symmetry: For all $x,y\in \mathcal{X}$, $x{\perp }^{T}y$ if and only if $y{\perp }^{T}x$.

(iv) 

Linear independence: For all $x,y\in \mathcal{X}-\left\{\mathbf{0}\right\}$, if $x{\perp }^{T}y$, then x and y are linearly independent.

 (v) 

Left additivity: For all $x_1,x_2,y\in \mathcal{X}$, if $x_1{\perp }^{T}y$ and $x_2{\perp }^{T}y$ then $x_1+x_2{\perp }^{T}y$.

Right additivity: For all $x,y_1,y_2\in \mathcal{X}$, if $x{\perp }^T{y}_1$ and $x{\perp }^T{y}_2$ then $x{\perp }^T{y}_1+y_2$.

(vi) 

Homogeneity: For all $x,y\in \mathcal{X}$, if $x{\perp }^{T}y$, then $\alpha x{\perp }^T\beta y$ for all $\alpha ,\beta \in \mathbb{C}$.

Proof. 

Items (i) and (ii) can be directly derived from properties (b) and (a) of Proposition 3, respectively.

For item (iii), evidently, equalities in (19) are symmetric for x and y. Moreover,

$$\begin{array}{cc}\hfill & G_2(x,iy,z)=0,\forall z\in \mathcal{X}\hfill \\ \hfill & \iff G_2(\frac{1}{i}x,y,z)=0,\forall z\in \mathcal{X}(from (a) of Lemma 11)\hfill \\ \hfill & \iff G_2(-ix,y,z)=0,\forall z\in \mathcal{X}\left(from\left(1\right)\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \iff G_2(ix,y,z)=0,\forall z\in \mathcal{X}.(from (c.2) ofLemma 11)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \iff G_2(y,ix,z)=0,\forall z\in \mathcal{X}.(from (c) ofProposition 3)\hfill \end{array}$$

(22)

The combination of (19), (20), and (22) indicates that property (iii) is true.

To prove item (iv), given $x,y\in \mathcal{X}-\left\{\mathbf{0}\right\}$, assume that $x{\perp }^{T}y$ and they would be linearly dependent, i.e.,

$$y=\alpha x,\alpha \in \mathbb{C},x,y\in \mathcal{X}-\left\{\mathbf{0}\right\}.$$

(23)

Substituting (23) into (19) and (20) yields

$${\parallel x+\alpha x+z\parallel }^2+{\parallel z\parallel }^2{-\parallel x+z\parallel }^2-{\parallel \alpha x+z\parallel }^2=0,\forall z\in \mathcal{X}$$

(24)

and

$${\parallel x+i\alpha x+z\parallel }^2+{\parallel z\parallel }^2{-\parallel x+z\parallel }^2-{\parallel i\alpha x+z\parallel }^2=0,\forall z\in \mathcal{X},$$

(25)

respectively. Taking $z=\mathbf{0}$ in (24) and (25), we obtain

$${|1+\alpha |}^2-1-{\left|\alpha \right|}^2=0$$

and

$${|1+i\alpha |}^2-1-{\left|\alpha \right|}^2=0,$$

which yield $Re\left(\alpha \right)=0$ and $Im\left(\alpha \right)=0$, respectively. This leads to $y=\mathbf{0}$ and contradicts the assumption (23).

Item (v) can be derived from equalities (19), (20), and property (b) of Lemma 11.

For item (vi), we note that, from property (c.1) of Lemma 11, condition (19) is equivalent to

$$G_2(ix,iy,z)=0,\forall z\in \mathcal{X}.$$

(26)

So equalities (19), (20), (21), and (26) all hold when $x{\perp }^{T}y$. For any given $\alpha =a+bi$, $\beta =c+di\in \mathbb{C}$, where $a,b,c,d\in \mathbb{R}$, from (19), (21), and property (c.3) of Lemma 11, we have $G_2(ax,y,z)=0$, $G_2(bix,y,z)=0$, $\forall z\in \mathcal{X}$. Then, according to property (b) of Lemma 11, we get

$$G_2(\alpha x,y,z)=0,\forall z\in \mathcal{X}.$$

(27)

Similarly, from (20) and (26), we have

$$G_2(\alpha x,iy,z)=0,\forall z\in \mathcal{X}.$$

(28)

Combining (27) and (28), and from properties (c.3) and (b) of Lemma 11, we obtain $G_2(\alpha x,\beta y,z)=0,\forall z\in \mathcal{X}$, and complete this proof. □

Remark 1. 

Condition (20) is obviously important for property (vi), and it also seems to be indispensable for property (iv), as evidenced by the derivation process around (25).

3.3. Comparison Between T-Orthogonality and Some Other Types of Orthogonality

Based on the above discussion, we can compare the T-orthogonality with several other classical orthogonality, which have been introduced in Definition 5.

Proposition 13. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a complex normed linear space and $x,y\in \mathcal{X}$. If $x{\perp }^{T}y$, then $x{\perp }_{P}y$, i.e.,

$${\parallel x+y\parallel }^2={\parallel x\parallel }^2+{\parallel y\parallel }^2.$$

(29)

Proof. 

Directly taking $z=\mathbf{0}$ in (19) hence results in the inequality (29). □

Proposition 14. 

Let $(\mathcal{X},||·\parallel )$ be a complex normed linear space and $x,y\in \mathcal{X}$. If $x{\perp }^{T}y$, then $x{\perp }_{R}y$, i.e., the equality

$$\parallel x+\alpha y\parallel =\parallel x-\alpha y\parallel$$

(30)

holds for any $\alpha \in \mathbb{C}$.

Proof. 

Under the assumption $x{\perp }^{T}y$ and according to the homogeneity of the T-orthogonality (the property (vi) of Proposition 12), for any given $\alpha \in \mathbb{C}$, we see that $x{\perp }^T(\pm \alpha )y$. Hence, from Proposition 13, it follows that

$${\parallel x+\alpha y\parallel }^2={\parallel x\parallel }^2+{\parallel \alpha y\parallel }^2={\parallel x-\alpha y\parallel }^2.$$

□

Corollary 15. 

Let $(\mathcal{X},||·\parallel )$ be a complex normed linear space and $x,y\in \mathcal{X}$. If $x{\perp }^{T}y$, then $x{\perp }_{I}y$, i.e.,

$$\left\{\begin{array}{c}\parallel x+y\parallel =\parallel x-y\parallel ,\\ \parallel x+iy\parallel =\parallel x-iy\parallel .\end{array}\right.$$

Moreover, if equality (30) holds for any $\alpha \in \mathbb{C}$, then (see also, e.g., [14,16])

$$2\parallel x\parallel =\parallel 2x+\alpha y-\alpha y\parallel \le \parallel x+\alpha y\parallel +\parallel x-\alpha y\parallel =2\parallel x+\alpha y\parallel .$$

(31)

Thus, from (31) and with be help of Proposition 14, we can also obtain the relation between T-orthogonality and Birkhoff orthogonality, as follows.

Corollary 16. 

Let $(\mathcal{X},||·\parallel )$ be a complex normed linear space and $x,y\in \mathcal{X}$. If $x{\perp }^{T}y$, then $x{\perp }_{B}y$, i.e.,

$$\parallel x+\alpha y\parallel \ge \parallel x\parallel$$

holds true for any $\alpha \in \mathbb{C}$.

Remark 2. 

The above results demonstrate that T-orthogonality is stronger than some other types of orthogonality. In particular, as far as we know, the fact that Proposition 14 reveals that T-orthogonality implies Roberts orthogonality is a new result.

3.4. Existence of Complex Normed Linear Spaces with T-Orthogonal Vectors as (Or Not) Inner Product Spaces

The following issue is a widely discussed topic:

Question 2. 

When can a normed space with (some type of) orthogonal vectors become an i.p.s.?

• Some results for T-orthogonality in complex normed linear spaces that are similar to those in [9] (in real normed linear spaces) still seem to hold, so we have to briefly state them in the last part of this section.

Proposition 17. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a complex normed linear space. If $x,y\in \mathcal{X}-\left\{\mathbf{0}\right\}$ and $x{\perp }^{T}y$, then $Span\{x,y\}$ is a c.i.p.s.

Proof. 

Under the assumption of this proposition, properties (iv) and (vi) of Proposition 12 tell us that x and y are independent and $\alpha x{\perp }^T\beta y$ for all $\alpha ,\beta \in \mathbb{C}$. According to Proposition 13, we have

$${\parallel \alpha x+\beta y\parallel }^2={\left|\alpha \right|}^2{\parallel x\parallel }^2+{\left|\beta \right|}^2{\parallel y\parallel }^2,\forall \alpha ,\beta \in \mathbb{C}.$$

(32)

Let arbitrary vectors $u,v\in Span\{x,y\}$ such that $u=c_{1}x+c_{2}y$ and $v=d_{1}x+d_{2}y$ with $c_1,c_2,d_2,d_2\in \mathbb{C}$. Then, from (32), it follows that

$$\begin{array}{cc}\hfill & {\parallel u+v\parallel }^2+{\parallel u-v\parallel }^2\hfill \\ \hfill & =\parallel (c_1+d_1)x+(c_2+d_2){y\parallel }^2+{\parallel (c_1-d_1)x+(c_2-d_2)y\parallel }^2\hfill \\ \hfill & \stackrel{\left(32\right)}{=}\left(|c_1+d_1{|}^2+{|c_1-d_1|}^2\right){\parallel x\parallel }^2+\left(|c_2+d_2{|}^2+{|c_2-d_2|}^2\right){\parallel y\parallel }^2\hfill \\ \hfill & =2|c_1{|}^2{\parallel x\parallel }^2+2|d_1{|}^2{\parallel x\parallel }^2+2|c_2{|}^2{\parallel y\parallel }^2+2|d_2{|}^2{\parallel y\parallel }^2\hfill \\ \hfill & \stackrel{\left(32\right)}{=}2\parallel c_{1}x+c_2{y\parallel }^2+2\parallel d_{1}x+d_2{y\parallel }^2={2\parallel u\parallel }^2+2{\parallel v\parallel }^2.\hfill \end{array}$$

Therefore, by Theorem 2, we conclude that $Span\{x,y\}$ is a c.i.p.s. □

Corollary 18. 

Let $(\mathcal{X},\parallel ·\parallel )$ be a two-dimensional complex normed linear space. Then, $\mathcal{X}$ is a c.i.p.s. if and only if there exist a couple of T-orthogonal vectors.

Corollary 18 implies that the T-orthogonality may sometimes be very close to the usual orthogonality (defined in Definition 3), especially in two-dimensional normed linear spaces. However, we have to note that the situation is different for more general spaces with dimensions greater than 2. This can be illustrated by the following Example 1. This example mainly refers to an example in [9] (p. 239), which was originally aimed at a real normed linear space, but we have now found that it can be modified to focus on a complex normed linear space.

Example 1. 

Consider the following map:

$$\begin{array}{cc}\hfill N_{\Phi ,\Psi }:{\mathbb{C}}^3& \to \mathbb{R}\hfill \\ \hfill \left(\begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\end{array}\right)& \mapsto N_{\Phi ,\Psi }({\xi }_1,{\xi }_2,{\xi }_3)={\left(\sum _{k=1}^3{\left|{\xi }_k\right|}^2+{∥\left(\begin{array}{c}{\xi }_1\\ {\xi }_3\end{array}\right)∥}_{\Phi }^2+{∥\left(\begin{array}{c}{\xi }_2\\ {\xi }_3\end{array}\right)∥}_{\Psi }^2\right)}^{1/2},\hfill \end{array}$$

where ${\parallel ·\parallel }_{\Phi }$ and ${\parallel ·\parallel }_{\Psi }$ denote two arbitrary given norms on ${\mathbb{C}}^2$. It is not difficult to verify that $N_{\Phi ,\Psi }$ is a norm on ${\mathbb{C}}^3$. Let $\mathcal{X}=({\mathbb{C}}^3,N_{\Phi ,\Psi })$, wherein let $e_i,i=1,2,3,$ be the canonical vectors and z be an arbitrary vector, with the following forms:

$$e_1=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),e_2=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),e_3=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),z=\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\end{array}\right).$$

It is easy to check that $G_2(e_1,e_2,z)=0$ and $G_2(e_1,i{e}_2,z)=0$ for all $z\in \mathcal{X}$, i.e., $e_1{\perp }^T{e}_2$. However, after some reduction, we obtain that

$$G_2(e_1,e_3,z)={∥\left(\begin{array}{c}1+z_1\\ 1+z_3\end{array}\right)∥}_{\Phi }^2+{∥\left(\begin{array}{c}{z}_1\\ z_3\end{array}\right)∥}_{\Phi }^2-{∥\left(\begin{array}{c}1+z_1\\ z_3\end{array}\right)∥}_{\Phi }^2-{∥\left(\begin{array}{c}{z}_1\\ 1+z_3\end{array}\right)∥}_{\Phi }^2,$$

$$G_2(e_1,i{e}_3,z)={∥\left(\begin{array}{c}1+z_1\\ i+z_3\end{array}\right)∥}_{\Phi }^2+{∥\left(\begin{array}{c}{z}_1\\ z_3\end{array}\right)∥}_{\Phi }^2-{∥\left(\begin{array}{c}1+z_1\\ z_3\end{array}\right)∥}_{\Phi }^2-{∥\left(\begin{array}{c}{z}_1\\ i+z_3\end{array}\right)∥}_{\Phi }^2,$$

$$G_2(e_2,e_3,z)={∥\left(\begin{array}{c}1+z_2\\ 1+z_3\end{array}\right)∥}_{\Psi }^2+{∥\left(\begin{array}{c}{z}_2\\ z_3\end{array}\right)∥}_{\Psi }^2-{∥\left(\begin{array}{c}1+z_2\\ z_3\end{array}\right)∥}_{\Psi }^2-{∥\left(\begin{array}{c}{z}_2\\ 1+z_3\end{array}\right)∥}_{\Psi }^2,$$

and

$$G_2(e_2,i{e}_3,z)={∥\left(\begin{array}{c}1+z_2\\ i+z_3\end{array}\right)∥}_{\Psi }^2+{∥\left(\begin{array}{c}{z}_2\\ z_3\end{array}\right)∥}_{\Psi }^2-{∥\left(\begin{array}{c}1+z_2\\ z_3\end{array}\right)∥}_{\Psi }^2-{∥\left(\begin{array}{c}{z}_2\\ i+z_3\end{array}\right)∥}_{\Psi }^2,$$

which indicate that $e_1$ will not be T-orthogonal to $e_3$ unless ${\parallel ·\parallel }_{\Phi }$ is an Euclidean norm (or derived from a special inner product) on ${\mathbb{C}}^2$, and $e_2$ will not be T-orthogonal to $e_3$ unless ${\parallel ·\parallel }_{\Psi }$ is an Euclidean norm (or derived from a special inner product) on ${\mathbb{C}}^2$.

Furthermore, we see that although there are a couple of T-orthogonal vectors $e_1,e_2$ in $\mathcal{X}$, the norm $N_{\Phi ,\Psi }$ cannot be derived from an inner product unless both ${\parallel ·\parallel }_{\Phi }$ and ${\parallel ·\parallel }_{\Psi }$ are derived from an inner product (readers can refer to [9] for detailed argumentation ideas, which are still applicable to the situation in ${\mathbb{C}}^3$). This shows the existence of complex normed linear spaces with T-orthogonal vectors that are not i.p.s.

4. Summary and Outlook

In this article, we extend the T-orthogonality from real normed linear spaces to complex normed linear spaces. The newly added condition (20) plays an important role in maintaining the linear dependence (property (iv)) and homogeneity (property (vi)) of T-orthogonality. Based on the study of some fundamental characterizations, we obtain that T-orthogonality implies Roberts orthogonality.

The issues that we currently lack research on but are interested in are the following:

• Can a normed linear space $\mathcal{X}$ be an i.p.s. if for all $x,y\in \mathcal{X}$, $x{\perp }_{R}y\Rightarrow x{\perp }^{T}y$?
• Can inequalities or equalities involving $G_p(x,y,z)$ with $p\ne 2$ also be applied to characterize orthogonality or other behaviors of some spaces?