Linear operators

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Problem: Will the operator \( (A x)(t)=\int_{0}^{1} \sin (t x(s)) d s \) contracting in the space \( C([0 ; 1]) \) with respect to the uniform metric?

19.3.1 Linear operators

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Problem: Find all the values of the parameter \( \alpha>0 \), for which the operator \( T: L^{2}[0 ; 1] \rightarrow L^{2}[0 ; 1] \), \( (T f)(x)=x f(x)-f\left(x^{\alpha}\right) \) is continuous.

19.3.2 Linear operators

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Problem: Find the spectrum of the operator \( A: l_{2} \rightarrow l_{2} \), \( A\left(x_{1}, x_{2}, \ldots, x_{n}, \ldots\right)=\left(x_{1}, 0, x_{2}, 0, \ldots, x_{n}, 0, \ldots\right) \).

19.3.3 Linear operators

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Problem: Prove the linearity and calculate the norm of the functional \( f(x)=\int_{0}^{2} t^{2} x(t) d t \), where a) \( x \in L_{1}[0,2] \), b) \( x \in L_{2}[0,2] \), c) \( x \in C[0,2] \).

19.3.4 Linear operators

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Problem: Prove the linearity and calculate the norm of the operator \( (A x)(t)=x(t) \), where a) \( A: C^{(1)}[0 ; 1] \rightarrow C[0,1] \), b) \( A: C[0,1] \rightarrow L_{1}[0,1] \).

19.3.5 Linear operators

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Problem: Prove that the sequence of operators \[ A_{n} x=(\underbrace{0, \ldots, 0}_{n}, x_{1}, \ldots, x_{n}, 0,0, \ldots) \] where \( A_{n}: l_{1} \rightarrow l_{1} \), doesn't converge pointwise to the zero operator.

19.3.6 Linear operators

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Problem: Find the kernel, image and the inverse operator for the operator \[ A x=\left(2 x_{1},-3 x_{2}, x_{3}, x_{4}, \ldots\right), A: l_{2} \rightarrow l_{2} \text {. } \]

19.3.7 Linear operators

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Problem: Prove that the functional \[ f: C[0,2] \rightarrow \mathbb{R} ; f(x)=\int_{0}^{2}(t-1)^{2} x(t) d t \] is linear, continuous, and find its norm.

19.3.8 Linear operators

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Problem: Find the norm of the function \( y=\tan x \) in the space \( L_{3}\left[\frac{\pi}{4}, \frac{\pi}{3}\right] \).

19.3.9 Linear operators

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