private void filterbox_TextChanged(object sender, EventArgs e)
{
    listView1.Items.Clear();   // clear all items we have atm
    if (filterbox.Text == "")
    {
        listView1.Items.AddRange(allItems.ToArray());  // no filter: add all items
        return;
    }
    // now we find all items that have a suitable text in any subitem/field/column
    var list = allItems.Cast()
                       .Where( x => x.SubItems
                                     .Cast()
                                     .Any(y => y.Text.Contains(filterbox.Text)))
                       .ToArray();
    listView1.Items.AddRange(list);  // now we add the result
}