To begin, you can simplify version with Predicate:

Predicate<object> firstName = pers =>
{
    var person = pers as Person;
    return person.Firstname.StartsWith(searchTerm); 
};

Predicate<object> lastName = pers => 
{
    var person = pers as Person;
    return person.Lastname.StartsWith(searchTerm); 
};

then you can replace with Func:

Func<object, bool> firstName = pers =>
{
    var person = pers as Person;
    return person.Firstname.StartsWith(searchTerm); 
};

Func<object, bool> lastName = pers =>
{
    var person = pers as Person;
    return person.Lastname.StartsWith(searchTerm); 
};

or use local functions (C# version >= 7.0):

bool firstName(object pers) 
{
    var person = pers as Person;
    return person.Firstname.StartsWith(searchTerm); 
};

bool lastName(object pers) 
{
    var person = pers as Person;
    return person.Lastname.StartsWith(searchTerm); 
};

or use own delegate:

public delegate bool SetName(object pers);
//...

SetName firstName = pers => 
{
    var person = pers as Person;
    return person.Firstname.StartsWith(searchTerm); 
};

SetName lastName = pers =>  
{
    var person = pers as Person;
    return person.Lastname.StartsWith(searchTerm); 
};

You're code seems to be fine, but we can rewrite it like below:

var Customer=(Customer)pers ;

Predicate<Customer> predict=x=>
x.Firstname.StartsWith(searchTerm) && x.Lastname.StartsWith(searchTerm);

Assuming, that:

• your question is about ICollectionView.Filter
• and you need to search using both first name and last name
• and Firstname and/or Lastname can be null

your code could be rewritten like this:

var searchTerm = search_box.Text;

yourCollectionView.Filter = obj =>
    obj is Person p &&
    (p.Firstname?.StartsWith(searchTerm) == true || 
     p.Lastname?.StartsWith(searchTerm) == true);

This, of course, uses Predicate<object>, since Filter property declared to be Predicate<object>, but you don't declare it explicitly.