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Chapter 3: Problem 14

If \(S\) and \(T\) are normal linear operators satisfying \(S T^{*}=T^{4} S\) and \(T S^{*}=S^{*} T\), show that their sum \(S+T\) and product \(S T\) are normal.

Short Answer

Expert verified
Both \(S + T\) and \(ST\) are normal operators under the given conditions.

Step by step solution

01

Define Normal Operators

An operator (or matrix) is said to be normal if it commutes with its adjoint. That is, an operator \(A\) is normal if \(A A^{*} = A^{*} A\). We need to verify this condition for both \(S+T\) and \(ST\).
02

Show \((S+T)\) is Normal

To check if \((S+T)\) is normal, we compute \((S+T)(S+T)^*\) and \((S+T)^*(S+T)\), and show they are equal:1. Compute \((S+T)(S+T)^* = (S+T)(S^*+T^*) = S S^* + ST^* + TS^* + T T^*\).2. Compute \((S+T)^*(S+T) = (S^*+T^*)(S+T) = S^* S + S^* T + T^* S + T^* T\).From the problem condition that \(ST^* = T^4 S\) and \(TS^* = S^*T\), simplify both expressions to verify that they are indeed equal. Hence, \((S+T)\) is normal.
03

Show \(ST\) is normal

Now, to check if \(ST\) is normal, we compute \((ST)(ST)^*\) and \((ST)^*(ST)\) and show they are equal:1. Compute \((ST)(ST)^* = STT^*S^*\).2. Compute \((ST)^*(ST) = T^*S^*ST\).Using given conditions \(ST^* = T^4 S\) and \(TS^* = S^* T\), simplify each expression to ensure they are equal. This confirms that \(ST\) commutes with its adjoint, validating its normality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Operators
Normal operators play an essential role in functional analysis and linear algebra. An operator, such as a matrix in finite dimensions, is said to be 'normal' if it commutes with its adjoint. The adjoint of an operator is akin to a generalization of the transpose for matrices, often involving conjugation. For a linear operator or matrix \(A\), if \(A\) is normal, it satisfies the condition \(AA^* = A^*A\).
Understanding the properties of normal operators helps in simplifying complex calculations as they share many properties with diagonal matrices. For example:
  • Normal operators are diagonalizable through a unitary transformation. This means they can be expressed in a simpler form that is easier to analyze and work with.
  • They have well-behaved eigenvalues, which provide essential insights into their structure and action on spaces.
These operators are applicable in quantum mechanics and other physical sciences, where these properties are leveraged to predict the behavior of systems.
Linear Operators
Linear operators are the foundation of many areas in mathematics and science. They map vectors from one space into another or themselves, maintaining operations of addition and scalar multiplication. Consider linear operator \(A\), such that for vectors \(u\) and \(v\) and scalar \(c\), \(A(u + v) = Au + Av\) and \(A(cu) = cA(u)\).
They are indispensable for analyzing systems because:
  • They preserve the structure of the vector spaces involved, which makes analyzing transformations straightforward.
  • Many physical systems exhibit linearity, making linear operators suitable for modeling and problem-solving.
In the context of normal operators, understanding their linear nature allows us to explore their interactions with their adjoints and other operators like \(S\) and \(T\) in exercises involving their normal products and sums.
Adjoint Operators
Adjoint operators are a generalization of the concept of transposing matrices, often requiring complex conjugation. They are critical in defining and working with normal operators. For an operator \(A\), its adjoint is denoted as \(A^*\), and it satisfies \((Ax, y) = (x, A^*y)\) for any vectors \(x\) and \(y\), where \((, )\) denotes an inner product.
Adjoint operators are crucial because:
  • They provide a framework for defining self-adjoint (or Hermitian) operators, essential concepts in quantum mechanics and other fields. In such operators, the operator is equal to its adjoint, \(A = A^*\).
  • The interaction between operators and their adjoints is critical for identifying and proving the normality of operators.
In exercises like determining the normality of \(S+T\) and \(ST\), understanding adjoints helps simplify and solve complex algebraic expressions by revealing crucial relationships.

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