List questions from compact-operators

Published on 18 Jun 2023 - 17:35

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Brezis' exercise 6.8: a compact operator with closed range is a finite-rank operator

Published on 19 Jun 2023 - 14:22

25

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Fredholm operator's measurablility derivation

Published on 20 Jun 2023 - 9:56

67

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Brezis' exercise 6.9.1: conditions equivalent to $\dim N(T) < \infty$ and $R(T)$ being closed

Published on 20 Jun 2023 - 12:07

24

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Brezis' exercise 6.9.2: $(T+S)$ satisfies (A) for every $S \in \mathcal K(E, F)$

Published on 20 Jun 2023 - 13:58

17

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Compact operator on a orthonormal sequence

Published on 26 Mar 2026 - 22:13

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Brezis' exercise 6.9.4: is there a shorter proof that $R(I+S)$ is closed?

Published on 21 Jun 2023 - 1:59

80

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Is this generalization of Fredholm alternative true?

Published on 21 Jun 2023 - 2:21

48

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Brezis' exercise 6.10.1: how to prove that $R(I-T)$ is closed?

Published on 23 Jun 2023 - 13:15

29

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Brezis' exercise 6.10.2: prove that $N(I-T) = \{0\} \iff R(I-T) = E$

Published on 24 Jun 2023 - 21:19

48

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Brezis' exercise 6.10.2: how does the author obtain $\left\|Q(T) u_n-Q(T) u_m\right\| \geq|Q(1)| / 2$?

Published on 24 Jun 2023 - 21:25

43

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Brezis' exercise 6.10.3: $\dim N(I-T) = \dim N(I-T^*)$

Published on 25 Jun 2023 - 15:04

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Show that the operator $Ax(t)=\int_0^1 \frac{\cos(t\tau)}{|t-\tau|^\frac{1}{5}}x(\tau)d\tau$ is compact on $L_2[0,1]$.

Published on 26 Jun 2023 - 19:35

61

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Brezis' exercise 6.12: $\forall \varepsilon>0\,\exists C_\varepsilon > 0$ s.t. $\|u\|_\infty \le\varepsilon \|u'\|_\infty + C_\varepsilon \|u\|_{L^1}$

Published on 26 Jun 2023 - 20:35

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Brezis' exercise 6.12: a lemma of J.-L. Lions

Published on 26 Jun 2023 - 22:28

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Questions regarding essentially normal operators in BDF theory

Brezis' exercise 6.8: a compact operator with closed range is a finite-rank operator

Fredholm operator's measurablility derivation

Brezis' exercise 6.9.1: conditions equivalent to $\dim N(T) < \infty$ and $R(T)$ being closed

Brezis' exercise 6.9.2: $(T+S)$ satisfies (A) for every $S \in \mathcal K(E, F)$

Compact operator on a orthonormal sequence

Brezis' exercise 6.9.4: is there a shorter proof that $R(I+S)$ is closed?

Is this generalization of Fredholm alternative true?

Brezis' exercise 6.10.1: how to prove that $R(I-T)$ is closed?

Brezis' exercise 6.10.2: prove that $N(I-T) = \{0\} \iff R(I-T) = E$

Brezis' exercise 6.10.2: how does the author obtain $\left\|Q(T) u_n-Q(T) u_m\right\| \geq|Q(1)| / 2$?

Brezis' exercise 6.10.3: $\dim N(I-T) = \dim N(I-T^*)$

Show that the operator $Ax(t)=\int_0^1 \frac{\cos(t\tau)}{|t-\tau|^\frac{1}{5}}x(\tau)d\tau$ is compact on $L_2[0,1]$.

Brezis' exercise 6.12: $\forall \varepsilon>0\,\exists C_\varepsilon > 0$ s.t. $\|u\|_\infty \le\varepsilon \|u'\|_\infty + C_\varepsilon \|u\|_{L^1}$

Brezis' exercise 6.12: a lemma of J.-L. Lions

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