What Does Self-Adjointness in Linear Operators Mean?

[yifli]

It is known that T in Hom(V) is self-adjoint if (Ta,b)=(a,Tb)

Let \theta be the isomorphism from V to V*, then (Ta, b)=(a,Tb) can be written as

$$\theta_b(Ta)=\theta_{Tb}(a)\rightarrow (T^*(\theta_b))(a)=\theta_{Tb}(a)\rightarrow T^*\circ \theta = \theta \circ T \rightarrow \theta^{-1} \circ T^* \circ \theta=T$$
where T* is the adjoint of T

So T is self-adjont means through certain transformation, T* can be transformed to T itself.

Is my interpretation of self-adjointness correct?

 

[jvicens]

Your interpretation of self-adjointness is correct. The concept of self-adjointness is a fundamental property of linear operators in a vector space. It means that an operator is equal to its own adjoint, or in other words, it is symmetric with respect to the inner product defined on the vector space. This property is important because it allows for the simplification of many mathematical calculations and also has important physical implications in quantum mechanics and other areas of physics. Your explanation using the isomorphism \theta is a clear and concise way of understanding this property.